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ffARREN  COLBUEN'S  FmST 


INTELLECTUAL  ARITHMETIC, 


INDUCTIVE  METHOD 


INSTRUCTION. 


BY  WARREN   COLBURN,  A.  M. 


WITH  AN 

INTRODUCTION  TO  WRITTEN  ARITHMETIC  BY  HIS  SON, 

WARREN  COLBURN,  AND  AN    INTRODUCTION 

BY  GEORGE  B.  EMERSON. 


PUBLISHED  FOR  THE  HEIRS  OF.  WARREN  COLBURN 

BY 

HURD  AND  HOUGHTON,  13  ASTOR  PLACE,,  NEW  YORK 
H.  H.  BANCROFT  &  CO.,  SAN  FRANCISCO,  CAL. 


OA 


JEntered  according  to  Act  of  Congress,  in  the  ye) 

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ID  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Masaachusetti 


Entered  according  to  Act  of  Congress,  in  the  year  1868,  by 

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Entered  according  to  Act  of  Congress  in  the  year  1863,  by 

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In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts 


APVERTISEMENT. 

fcDUcXrjoi   DI 

THIS  book  has  returned  by  purchase  into  the  pos 
session  of  the  heirs  of  the  late  WARREN  COLBURN, 
and  will  hereafter  be  published  for  their  benefit. 
The  Introduction,  by  GEORGE  B.  EMERSON,  gives 
some  valuable  hints  as  to  the  proper  use  of  the  book. 
The  Original  Preface,  in  which  the  design  and  plan 
of  the  work  are  stated  at  length  by  the  Author,  has 
been  restored.  A  careful  perusal  of  this  Introduction 
and  Preface  is  earnestly  recommended  to  all  teachers 
who  make  use  of  the  book. 

H.  0.  HOUGHTON. 

Uambridye,  June,  1863. 


INTRODUCTION. 


THIS  little  book  has  now  been  in  constant  use  foi 
forty  years,  and  its  value  has  been  proved  by  the 
entire  and  admiring  approbation  of  thousands  of  the  best 
teachers.  Yet  its  very  simplicity  has  prevented  mauy 
persons  from  seeing  how  really  profound  and  comprehen- 
sive it  is,  and  that  it  actually  develops  every  essential  prin- 
ciple in  elementary  arithmetic.  Want  of  attention  and 
the  previous  use  of  poorer  Books  have  also  misled  many 
persons  as  to  the  proper  and  true  mode  of  using  it. 

It  is  strictly  a  mental  arithmetic,  and,  if  faithfully  used 
in  the  way  intended  by  the  author,  it  evolves  from  the 
mind  of  the  learner  himself,  in  a  perfectly  easy  and  natural 
manner,  a  knowledge  of  the  principles  of  arithmetic,  and 
the  power  of  solving,  mentally  and  almost  instantly,  every 
question  likely  to  occur  in  the  every-day  business  of  com- 
mon life. 

It  can  be  well  taught  only  by  a  teacher  who  perfectly 
understands  it,  and  who  knows  how  to  teach.  Such  a 
teacher  will  not  allow  the  lesson  to  be  previously  studied 
by  the  pupil.  Each  section  is  intended  to  teach  some  one 
process  up  to  a  certain  point.  If,  in  the  course  of  the  sec- 
tion, questions  occur  which  the  class  cannot  readily  solve 
without  previous  study,  the  teacher  has  only  to  interpose, 
at  the  point  where  the  class  fails,  or  begins  to  fail,  addi- 

733 


4  INTRODUCTION. 

•"tibri&l  q&eatioW  b'f  'the  same  kind,  somewhat  easier  than 
those  in  the  book.  If,  at  the  end  of  the  section,  the  class 
be  not  perfectly  ready  in  the  solution  of  the  questions,  the 
teacher  ought  to  go  over  the  section  again,  with  the  class, 
or  to  add,  at  the  end  of  the  section,  a  sufficient  number  of 
similar  questions  to  render  the  solution  easy  and  instanta- 
neous. 

By  allowing  the  class  to  study  the  lesson  beforehand,  not 
only  is  much  time  lost,  but  the  exercise  is  turned  into  a 
poor  sort  of  mechanical  process  not  much  better  than  the 
common  ciphering.  Its  mental  character  ceases  almost  en- 
tirely. 

As  an  exercise  in  exact  reasoning,  the  mental  solution 
of  questions  is  very  valuable ;  not  less  so  than  geometrical 
demonstration,  and  preferable  in  the  fact  that  it  is  purely 
mental.  It  is  thus  an  admirable  preparation  for  the  attain- 
ment of  the  faculty  of  ready  and  sharp  logic,  to  be  applied 
to  the  business  of  common  life,  where  the  mental  habit 
alone  is  available.  Its  great  value  for  children  is  in  its 
shortness  and  its  clearness.  For  example  :  —  What  is  J.  ot 
|  of  |  of  |  ?  Answer :  £  of  §  of  £  of  £  is  £  ;  for  £  of  § 
*  A?  4  of  £  is  I  ;  i  of  t  *  1 5  therefore,*,  of  §  of  3  of  | 
is  L.  Similar  examples  are  found  jn  almost  every  section 
in  the  book.  Now,  the  going  faithfully  through  many  such 
processes,  every  day  for  years,  at  the  age  at  which  mental 
habits  are  formed,  must  have  the  effect  of  forming  habits 
of  clearness,  accuracy,  and  exact  reasoning,  which,  once 
fully  formed,  will  always  exert  a  happy  influence  over  the 
mind. 

This  book  ought  not  to  be  used  with  beginners  only.  Its 
effect  upon  those  far  advanced  is  even  better  and  more 
striking.  It  might  be  continued,  with  the  greatest  advan- 


INTRODUCTION/'  5 

tage,  till  the  end  of  every  course  of*  MlstVuciionJ  and*, 'ft*  »sc'  ' 
continued,  would  Lave  the  constant  effect  of  sharpening 
the  perception  and  varying  the  ability  of  analysis,  of 
giving  perfect  readiness  in  the  mental  solution  of  all  com- 
mon problems,  and  of  fixing  the  habit  of  clearness,  quick- 
ness, and  accuracy  in  the  process  of  exact  reasoning. 

GEORGE  B.  EMERSON. 
May,  1863. 


THE  ORIGINAL   PREFACE. 


As  soon  as  a  child  begins  to  use  his  senses,  nature  con- 
tinually presents  to  his  eyes  a  variety  of  objects ;  and  one  of 
the  first  properties  which  he  discovers  is  the  relation  of 
number  He  intuitively  fixes  upon  unity  as  a  measure  and 
from  this  he  forms  the  idea  of  more  and  less  ;  which  is  the 
idea  of  quantity. 

The  names  of  a  few  of  the  first  numbers  are  usually 
learned  very  early  ;  and  children  frequently  learn  to  count 
as  far  as  a  hundred  before  they  learn  their  letters. 

As  soon  as  children  have  the  idea  of  more  and  less,  and 
the  names  of  a  few  of  the  first  numbers,  they  are  able  to 
make  small  calculations.  And  this  we  see  them  do  every 
day  about  their  playthings,  and  about  the  little  affairs 
which  they  are  called  upon  to  attend  to.  The  idea  of  more 
and  less  implies  addition ;  hence  they  will  often  perform 
these  operations  without  any  previous  instruction.  If,  for 
example,  one  child  has  three  apples,  and  another  five,  they 
will  readily  tell  how  many  they  both  have  ;  and  how  many 
one  has  more  than  the  other.  If  a  child  be  requested  to 
bring  three  apples  for  each  person  in  the  room,  he  will  cal- 
culate very  readily  how  many  to  bring,  if  the  number  does 
not  exceed  those  he  has  learnt.  Again,  if  a  child  be  re- 
quested to  divide  a  number  of  apples  among  a  certain 
number  of  persons,  he  will  contrive  a  way  to  do  it,  and  will 


ORIGINAL   PREFACE. 


Lave.  The  method  which  chil- 
dren take  to  do  these  things,  though  always  correct,  it 
not  always  the  most  expeditious. 

The  fondness  which  children  usually  manifest  for  thew 
exercises,  and  the  facility  with  which  they  perform  them, 
seem  to  indicate  that  the  science  of  numbers,  to  a  certain 
extent,  should  be  among  the  first  lessons  taught  to  them. 

To  succeed  in  this,  however,  it  is  necessary  rather  to 
furnish  occasions  for  them  to  exercise  their  own  skill  in 
performing  examples,  than  to  give  them  rules.  Thoy 
should  oe  allowed  to  pursue  their  own  method  first,  and 
then  "they  should  be  made  to  observe  and  explain  it  ;  r.nd  if 
it  was  not  the  best,  some  improvement  should  !>••  Mi«;_in\<t»'d 
By  following  this  mode,  and  making  the  examples  gradu- 
ally increase  in  difficulty,  experience  proves,  that  at  an 
early  age,  children  may  be  taught  a  great  variety  of  tli<» 
most  useful  combinations  of  numbers. 

Few  exercises  strengthen  and  mature  the  mini  so  much 
as  arithmetical  calculations,  if  the  examples  are  made  suffi- 
ciently simple  to  be  understood  by  the  pupil  ;  because  a 
regular  though  simple  process  of  reasoning  is  requisite  to 
perform  them,  and  the  results  are  attended  with  certainty. 

The  idea  of  number  is  first  acquired  by  observing  sen- 
sible objects.  Having  observed  that  this  quality  is  common 
to  all  things  with  which  we  are  acquainted,  we  obtain  an 
abstract  idea  of  number.  We  first  make  calculations  about 
sensible  objects  ;  and  we  soon  observe  that  the  same  calcu- 
lations will  apply  to  things  very  dissimilar  ;  and  finally, 
that  they  may  be  made  without  reference  to  any  particular 
things.  Hence,  from  particulars,  we  establish  general  prin- 
ciples, which  serve  as  the  basis  of  our  reasonings,  and  en- 
able us  to  proceed,  step  by  step,  from  the  most  simple  to  the 


THE   ORIGINAL   PREFA'Cfc.-  '':/4^  '9H  ' 

more  complex  operations.  It  appeals,  ,t1i^ri 
mathematical  reasoning  proceeds  as  much  upon 
ciple  of  analytic  induction,  as  that  of  any  other  ali£Bc< 

From  the  above  observations,  and  from  his 
ence,  the  author  has  been  induced  to  publish  this 
in  which  he  has  pursued  the  following  plan,  which 
to  him  the  most  agreeable  to  the  natural  progress 
mind. 

GENERAL  VIEW  OF  THE  PLAN. 

EVERY  combination  commences  with  practical  exam- 
ples. Care  has  been  taken  to  select  such  as  will  aptly 
illustrate  the  combination,  and  assist  the  imagination  of 
the  pupil  in  performing  it.  In  most  instances,  immediately 
after  the  practical,  abstract  examples  are  placed,  contain- 
ing the  same  numbers  and  the  same  operations,  that  the 
pupil  may  the  more  easily  observe  the  connection.  The 
instructor  should  be  careful  to  make  the  pupil  observe  the 
connection.  After  these  are  a  few  abstract  examples,  and 
then  practical  questions  again. 

The  examples  are  to  be  performed  in  the  mind,  or  by 
means  of  sensible  objects,  such  as  beans,  nuts,  &c.  The 
pupil  should  first  perform  the  examples  in  his  own  way, 
and  then  be  made  to  observe  and  tell  how  he  <3H  them, 
and  why  he  did  them  so.* 

*  It  is  remarkable  that  a  child,  although  he  is  able  to  perform 
a  variety  of  examples  which  involve  addition,  subtraction,  multi- 
plication, and  division,  recognizes  no  operation  but  addition.  In 
deed,  if  we  analyze  these  operations  when  we  perform  them  ix 
our  minds,  we  shall  find  that  they  all  reduce  themselves  to  addi 
tion.  They  are  only  different  ways  of  applying  the  same  princk 
ole.  And  it  is  only  when  we  use  an  artificial  method  of  perform- 
5ng  them,  thaf  they  take  a  different  form. 


10  v    •  :•'  JIBE   ORIGINAL   PREFACE. 
•  •  V  .Tht?  ijrst  jscHitiqn  .contains  addition  and  subtraction,  the 


second  multiplication.  The  third  section  contains  division 
In  this  section  the  pupil  learns  the  first  principles  of  frac- 
tions and  the  terms  which  are  applied  to  them.  This  is 
done  by  making  him  observe,  that  one  is  the  half  of  two, 
the  third  of  three,  the  fourth  of  four,  &c.,  and  that  two  is 
two  thirds  of  three,  two  fourths  of  four,  two  fifths  of  five, 
&c. 

The  fourth  section  commences  with  multiplication.  In 
this  the  pupil  is  taught  to  repeat  a  number  a  certain  num- 
ber of  times,  and  a  part  of  another  time.  In  the  second 
part  of  this  section  the  pupil  is  taught  to  change  a  certain 
ouniber  of  twos  into  threes,  threes  into  fours,  &c. 

If  the  following  questions  were  proposed  to  a  child,  his  answers 
would  be,  in  substance,  like  those  annexed  to  the  questions.  How 
much  is  five  less  than  eight  ?  Ans.  :  Three.  Why  ?  Because  five 
and  three  are  eight.  What  is  the  difference  between  five  and 
eight?  Ans.:  Three.  Why?  Because  five  and  three  are  eight. 
If  you  divide  eight  into  two  parts,  such  that  one  of  the  parts  may 
be  five,  what  will  the  other  be?  Ans.:  Three.  Why?  Because 
five  and  three  are  eight. 

How  much  must  you  give  for  four  apples  at  two  cents  apiece? 
Ans.  :  Eight  cents.  Why  ?  Because  two  and  two  are  four,  and  two 
are  six,  and  two  are  eight. 

How  many  apples,  at  two  cents  apiece,  can  you  buy  for  eight 
cents  ?  Ans.  :  Four.  Why  ?  Because  two  and  two  are  four,  and 
two  are  six,  and  two  are  eight. 

We  shall  be  further  convinced  of  this,  if  we  observe  that  the 
same  table  serves  for  addition  and  subtraction  ;  and  another  table, 
which  is  formed  by  addition,  serves  both  for  multiplication  and  di- 
vision. 

This  remark  shows  the  necessity  of  making  the  pupil  attend  to 
his  manner  of  performing  the  examples  and  of  explaining  to  him 
the  difference  between  them. 


THE  ORIGINAL  PREFACE 


11 


In  the  fifth  section  the  pupil  is  taugHt,*o;fiA$  |, 
And  g,  |,  |,  &c.,  of  numbers  which  are  exactly  divisible 
into  these  parts.     This  is  only  an  extension  of  the  principle  ~ 
of  fractions,  which  is  contained  in  the  third  section. 

In  the  sixth  section  the  pupil  learns  to  tell  of  what  num- 
ber any  number,  as  2,  3>  4,  &c.,  is  one  half,  one  third,  one 
fourth,  &c. ;  and  also  knowing  J,  |?  4,  &c.  of  a  number,  to 
find  that  number. 

.  These  combinations  contain  all  the  most  common  and 
most  useful  operations  of  vulgar  fractions.  But  being  ap- 
plied only  to  numbers  which  are  exactly  divisible  into  these 
fractional  parts,  the  pupil  will  observe  no  principles  but 
multiplication  and  division,  unless  he  is  told  of  it.  In  fact, 
fractions  contain  no  other  principle.  The  examples  are  so 
arranged  that  almost  any  child  of  six  or  seven  years  old 
will  readily  comprehend  them.  And  the  questions  are 
asked  in  such  a  manner  that,  if  the  instructor  pursues  the 
method  explained  in  the  Key,  it  will  be  almost  impossible 
for  the  pupil  to  perform  any  example  without  understand- 
ing the  reason  of  it.  Indeed,  in  every  example  which 
he  performs,  he  is  obliged  to  go  through  a  complete 
demonstration  of  the  principle  by  which  he  does  it ;  and 
at  the  same  time  he  does  it  in  the  simplest  way  possible. 
These  observations  apply  to  the  remaining  part  of  the 
book. 

These  principles  are  sufficient  to  enable  the  pupil  to 
perform  almost  all  kinds  of  examples  that  ever  occur.  He 
will  not,  however,  be  able  to  solve  questions  in  which  it  is 
necessary  to  take  fractional  parts  of  unity,  though  the 
principles  are  the  same. 

After  section  sixth,  there  is  a  collection  of  miscellaneous 
s,  in  which  are  contained  almost  all  the  kinds  that 


:  :'*Tk£    ORIGINAL   PREFACE. 


are  none,  however,  which  the  prin« 
ciples  explained  are  not  sufficient  to  solve. 

In  section  eighth  and  the  following,  fractions  of  unity  are 
explained,  and,  it  is  believed,  so  simply  as  to  be  intelligible 
to  most  pupils  of  seven  or  eight  years  of  age.  The  opera- 
tions do  not  differ  materially  from  those  in  the  preceding 
sections.  There  are  some  operations,  however,  peculiar  to 
fractions. 

When  the  pupil  is  made  familiar  with  all  the  principles 
contained  in  this  book,  he  will  be  able  to  perform  all  exam- 
ples, in  which  the  numbers  are  so  small  that  the  opera- 
tions may  be  'performed  in  the  mind.  Afterwards  he  has 
only  to  learn  the  application  of  figures  to  these  operations, 
and  his  knowledge  of  arithmetic  will  be  complete. 

The  following  are  some  of  the  principal  difficulties  which 
a  child  has  to  encounter  in  learning  arithmetic,  in  the 
usual  way,  and  which  are  seldom  overcome.  First,  the  ex- 
amples are  so  large  that  the  pupil  can  form  no  conception 
of  the  numbers  themselves  ;  therefore  it  is  impossible  for 
him  to  comprehend  the  reasoning  upon  them.  Secondly, 
the  first  examples  are  usually  abstract  numbers.  Thi? 
increases  the  difficulty  very  much  ;  for  even  if  the  numbers 
were  so  small  that  the  pupil  could  comprehend  them,  he 
would  discover  but  very  little  connection  between  them 
and  practical  examples.  Abstract  numbers,  and  the  op- 
erations upon  them,  must  be  learned  from  practical  exam- 
ples ;  there  is  no  such  thing  as  deriving  practical  examples 
from  those  which  are  abstract,  unless  the  abstract  have 
been  first  derived  from  those  which  are  practical.  Thirdly, 
the  numbers  are  expressed  by  figures,  which,  if  they  were 
used  only  as  a  contracted  way  of  writing  numbers,  would 
be  much  more  difficult  to  be  understood  at  first  than  the 


THE   ORIGINAL   PREFACE  13 


numbers  written  at  length  in  words.  ^  .But1  4;^y  ^  are  'nbt 
used  merely  as  words,  they  require  operations  peculiar  to 
themselves.  They  are,  in  fact,  a  new  language,  which  the 
pupil  has  to  learn.  The  pupil,  therefore,  when  he  com- 
mences arithmetic,  is  presented  with  a  set  of  abstract  num- 
bers, written  \f\\hfgures,  and  so  large  that  he  has  not  the 
least  conception  of  them  even  when  expressed  in  words. 
From  these  he  is  expected  to  learn  what  the  figures  sig- 
nify, and  'what  is  meant  by  addition,  subtraction,  multiplica- 
tion, and  division;  and  at  the  same  time  how  to  perform 
these  operations  with  figures.  The  consequence  is,  that  he 
learns  only  one  of  all  these  things,  and  that  is,  how  to  per- 
form these  operations  on  figures.  He  can  perhaps  translate 
the  figures  into  words,  but  this  is  useless,  since  he  does  not 
understand  the  words  themselves.  Of  the  effect  produced 
by  the  four  fundamental  operations  he  has  not  the  least 
conception. 

After  the  abstract  examples,  a  few  practical  examples 
are  usually  given,  but  these  again  are  so  large  that  the 
pupil  cannot  reason  upon  them,  and  consequently  he  could 
not  tell  whether  he  must  add,  subtract,  multiply,  or  divide, 
even  if  he  had  an  adequate  idea  of  what  these  operations 
are. 

The  common  method,  therefore,  entirely  reverses  the 
natural  process  ;  for  the  pupil  is  expected  to  learn  general 
principles,  before  he  has  obtained  the  particular  ideas  of 
which  they  are  composed. 

The  usual  mode  of  proceeding  is  as  follows.  The  pupil 
learns  a  rule,  which,  to  the  man  that  made  it,  was  a  gen- 
eral principle  ;  but  with  respect  to  him,  and  oftentimes  to 
the  instructor  himself,  it  is  so  far  from  it,  that  it  hardly  de- 
serves to  be  calkd  even  a  mechanical  principle.  He  per* 


14 


•'  THE   ORIGINAL   PKEFACF 


•fdrftis  ^tAe- exzftipkA,  and  makes  the  answers  agree  with 
those  in  the  book,  and  so  presumes  they  are  right.  He  la 
soon  able  to  do  this  with  considerable  facility,  and  is  then 
supposed  to  be  master  of  the.  rule.  He  is  next  to  apply 
his  rule  to  practical  examples ;  but  if  he  did  not  find  the 
examples  under  the  rule,  he  would  never  so  much  as  mis- 
trust they  belonged  to  it.  But  finding  them  there,  he 
applies  his  rule  to  them,  and  obtains  the  answers,  which 
are  in  the  book,  and  this  satisfies  him  that  they  are  right. 
In  this  manner  he  proceeds  from  rule  to  rule  through  the 
book. 

When  an  example  is  proposed  to  him,  which  is  not  in 
the  book,  his  sagacity  is  exercised,  not  in  discovering  the 
operations  necessary  to  solve  it,  but  in  comparing  it  with 
the  examples  which  he  has  performed  before,  and  endeav- 
oring to  discover  some  analogy  between  it  and  them, 
either  in  the  sound,  or  in  something  else.  If  he  is  fortu- 
nate enough  to  discover  any  such  analogy,  he  finds  what 
rule  to  apply ;  and  if  he  has  not  been  deceived  in  tracing 
the  analogy,  he  will  probably  solve  the  question.  His 
knowledge  of  the  principles  of  his  rules  is  so  imperfect,  that 
he  would  never  discover  to  which  of  them  the  example 
belongs,  if  he  did  not  trace  it,  by  some  analogy,  to  the 
examples  which  he  had  found  under  it. 

These  observations  do  not  apply  equally  to  all ;  for  some 
will  find  the  right  course  themselves,  whatever  obstacles  be 
thrown  in  their  way.  But  they  apply  to  the  greater  part ; 
and  it  is  probable  that  there  are  very  few  who  have  not 
experienced  more  or  less  inconvenience  from  this  mode  of 
proceeding.  Almost  all,  who  have  ever  fully  understood 
arithmetic,  have  been  obliged  to  learn  it  over  again  in  their 
own  way.  And  it  is  not  too  bold  an  assertion  to  say,  that 


THE  ORIGINAL  PREFACE.  15 

no  man  ever  actually  learned  mathematics  in  any  other 
method  than  by  analytic  induction ;  that  is,  by  teaming 
the  principles  by  the  examples  he  performs ;  and  not  by 
learning  principles  first,  and  then  discovering  by  them 
how  the  examples  are  to  be  performed. 


THE  BOY  WITHOUT  A  GENIUS. 

MR.  WISEMAN,  the  schoolmaster,  at  the  end  of  his  summer 
vacation,  received  a  new  scholar,  with  the  following  letter : 

SIR,  —  This  will  be  delivered  to  you  by  my  son,  Samuel, 
whom  I  beg  leave  to  commit  to  your  care,  hoping  that  by 
your  well-known  skill  and  attention  you  will  be  able  to  make 
something  of  him ;  which,  I  am  sorry  to  say,  none  of  his 
masters  have  hitherto  done.  He  is  now  eleven,  and  yet  can 
do  nothing  but  read  his  mother-tongue,  and  that  but  indiffer- 
ently. We  sent  him  at  seven  to  a  grammar  school  in  our 
neighborhood  ;  but  his  master  soon  found  that  his  genius  was 
not  turned  to  learning  languages.  He  was  then  put  to  writ- 
ing, but  he  set  about  it  so  awkwardly  that  he  made  nothing  of 
it.  He  was  tried  at  accounts,  but  it  appeared  that  he  had  no 
genius  for  that  neither.  He  cpuld  do  nothing  in  geography 
for  want  of  memory.  In  short,  if  he  has  any  genius  at  all, 
it  does  not  yet  show  itself.  But  I  trust  to  your  experience 
in  cases  of  this  nature  to  discover  what  he  is  fit  for,  and  to 
instruct  him  accordingly.  I  beg  to  be  favored  shortly  with 
your  opinion  about  him,  and  remain,  sir, 

Your  most  obedient  servant, 

HUMPHREY  ACRES. 

When  Mr.  Wiseman  had  read  this  letter,  he  shook  his  head, 
and  said  to  his  assistant,  A  pretty  "subject  they  have  sent  us 
here  !  a  lad  that  has  a  great  genius  for  nothing  at  all.  But 
perhaps  my  friend  Mr.  Acres  expects  that  a  boy  should  show 
a  genius  for  a  thing  before  he  knows  anything  about  it,  —  no 
uncommon  error!  Let  us  see,  however,  what  the  youth 
looks  like.  I  suppose  he  is  a  human  creature  at  least. 

Master  Samuel  Acres  was  now  called  in.  He  came,  hang- 
ing down  his  head,  and  looking  as  if  he  was  going  to  be 
flogged. 


16  THE   ORIGINAL   PREFACE. 

Come  hither,  my  dear !  said  Mr.  Wiseman.  Stand  by  me, 
and  do  not  be  afraid.  Nobody  will  hurt  you.  How  old  are 
you? 

Eleven  last  May,  sir. 

A  well-grown  boy  of  your  age,  indeed.  You  love  play,  I 
dare  say  ? 

Yes,  sir. 

What,  are  you  a  good  hand  at  marbles  1 

Pretty  good,  sir. 

And  can  spin  a  top  and  drive  a  hoop,  I  suppose  ? 

Yes,  sir. 

Then  you  have  the  full  use  of  your  hands  and  fingers  f 

Yes,  sir. 

Can  you  write,  Samuel  ? 

I  learned  it  a  little,  sir,  but  I  left  it  off  again. 

And  why  so  ? 

Because  I  could  not  make  the  letters. 

No  !  why,  how  do  you  think  other  boys  do  ?  Have  they 
more  fingers  than  you  ? 

No,  sir. 

Are  you  not  able  to  hold  a  pen  as  well  as  a  marble  ? 

Samuel  was  silent. 

Let  me  look  at  your  hand. 

I  see  nothing  here  to  hinder  you  from  writing  as  well  as 
any  boy  in  the  school.  You  can  read,  I  suppose  ? 

Yes,  sir. 

Tell  me,  then,  what  is  written  over  the  school-room  door  ? 

Samuel,  with  some  hesitation,  read,  WHATEVER  MAN  HAS 

DONE  MAN  MAY  DO. 

Pray,  how  did  you  learn  to  read  ?  Was  it  not  with  taking 
pains  ? 

Yes,  sir. 

Well,  —  taking  more  pains  will  enable  you  to  read  better. 
Do  you  know  anything  of  the  Latin  grammar  ? 

No,  sir. 

Have  you  never  learned  it  ? 

I  tried,  sir,  but  I  could  not  get  it  by  heart. 

Why,  you  can  say  some  things  by  heart.  I  dare  say  you 
can  tell  me  the  names  of  the  days  of  the  week  in  their  order 

Yes,  sir,  I  know  them. 

And  the  months  in  the  year,  perhaps  ? 

Yes,  sir. 

And  you  could  probably  repeat  the  names  of  your  brother? 
and  sisters,  and  all  your  father's  servants,  and  half  the  people 
in  the  village  besides  ? 

I  believe  I  qould,  sir 


THE   ORIGINAL  PREFACE.  17 

Well,  —  and  is  hie,  hcec,  hoc,  more  difficult  to  remember  than 
these  ? 

Samuel  was  silent. 

Have  you  Learned  a.  ything  of  accounts  ? 

I  went  into  addition  sir,  but  I  did  not  go  on  with  it. 

Why  so  ? 

I  could  not  do  it,  sir. 

How  many  marbles  can  you  buy  for  a  penny  ? 

Twelve  new  ones,  sir. 

And  how  many  for  a  half-penny  ? 

Six. 

And  how  many  for  two-pence  ? 

Twenty-four. 

If  you 'were  to  have  a  penny  a  day,  what  would  that  make 
in  a  week  ? 

Seven-pence. 

But  if  you  paid  two-pence  out  of  that,  what  would  you 
have  left  ? 

Samuel  studied  a  while,  and  then  said,  five-pence. 

Right.  Why,  here  you  have  been  practising  the  four  great 
rules  of  arithmetic,  —  addition,  subtraction,  multiplication, 
and  division.  Learning  accounts  is  no  more  than  this.  Well, 
Samuel,  I  see  what  you  are  fit  for.  I  shall  set  you  about 
nothing  but  what  you  are  able  to  do ;  but  observe,  you  must 
do  it.  We  have  no  I  can't  here.  Now,  go  among  your  school- 
fellows. 

Samuel  went  away,  glad  that  his  examination  was  over, 
and  with  more  confidence  in  his  powers  than  he  had  felt 
before. 

The  next  day  he  began  business.  A  boy  less  than  himself 
was  called  out  to  set  him  a  copy  of  letters,  and  another  was 
appointed  to  hear  him  in  grammar.  He  read  a  few  sentences 
hi  English,  that  he  could  perfectly  understand,  to  the  master 
himself.  Thus  by  going  on  steadily  and  slowly,  he  made  a 
sensible  progress.  He  had  already  joined  his  letters,  got  all 
the  declensions  perfectly,  and  half  the  multiplication-table, 
when  Mr.  Wiseman  thought  it  time  to  answer  his  father's 
letter;  whicfi  he  did  as  follows : 

SIR,  —  I  now  think  it  right  to  give  you  some  information 
Concerning  your  son.  You  perhaps  expected  it  sooner,  but 
[  always  wish  to  avoid  hasty  judgments.  You  mentioned  in 
your  letter  that  it  had  not  yet  been  discovered  which  way  his 
genius  pointed.  If  by  genius  you  meant  such  a  decided  bent 
of  mind  tc  any  one  pursuit  as  will  lead  to  excel  with  little  or 
no  labor  or  instruction,  I  must  say  that  I  have  not  met  with 


18  THE  ORIGINAL   PREFACE 

such  a  quality  in  more  than  three  or  four  boys  in  my  life,  and 
your  son  is  certainly  not  among  the  number.  But  if  you 
mean  only  the  ability  to  do  some  of  those  things  which  the 
greater  part  of  mankind  can  do  when  properly  taught,  I  can 
affirm  that  I  find  in  him  no  peculiar  dt  riciency.  And  whether 
you  choose  to  bring  him  up  to  trade  <  r  to  some  practical  pro- 
fession, I  see  no  reason  to  doubt  that  he  may  in  time  become 
sufficiently  qualified  for  it.  It  is  my  favorite  maxim,  sir, 
that  everything  most  valuable  in  this  life  may  generally  be 
acquired  by  taking  pains  for  it.  Your  son  has  already  lost 
much  time  in  the  fruitless  expectation  of  finding  out  what  he 
would  take  up  of  his  own  accord.  Believe  me,  sir,  few  boys 
will  take  up  anything  of  their  own  accord  but  a  top  or  a 
marble.  I  will  take  care,  while  he  is  with  me,  that  he 
loses  no  more  time  this  way,  but  is  employed  about  tilings 
that  are  fit  for  him,  not  doubting  that  we  shall  find  him  fit  for 
them.  I  am,  sir,  yours,  &c., 

SOLON  WISEMAN. 

Though  the  doctrine  of  this  letter  did  not  perfectly  agree 
frith  Mr.  Acres'  notions,  yet  being  convinced  that  Mr.  Wise- 
man was  more  likely  to  make  something  of  his  son  than  any 
of  his  former  preceptors,  he  continued  him  at  his  school  for 
some  years,  and  had  the  satisfaction  to  find  him  going  on  in 
a  steady  course  of  gradual  improvement.  In  due  time  a  pro- 
fession was  chosen  for  him,  which  seemed  to  suit  his  temper 
and  talents,  but  for  which  Re  had  no  particular  turn,  having 
never  thought  at  all  about  it.  He  made  a  respectable  figure 
in  it,  and  went  through  the  world  with  credit  and  usefulness, 
though  without  a  genius. 

Mrs.  Barbauld. 


ARITHMETIC. 
PART   I. 

SECTION  I. 

A.*  1.  How  many  thumbs  have  you  on  your  right 
hand?  how  many  on  your  left?  how  many  on  both 
together  ? 

2.  How  many  hands  have  you  ? 

3.  If  you  have  two  nuts  in  one  hand,  and  one  in 
the  other,  how  many  have  you  in  both  ? 

4.  How  many  fingers  have  you  on  one  hand  ? 

5.  If  you  count  the  thumb  with  the  fingers,  how 
many  will  it  make  ? 

6.  If  you  shut  your  thumb  and  one  finger,  and 
leave  the  rest  open,  how  many  will  be  open  ? 

7.  If  you  have  two  cents  in  one  hand,  and  two  in 
the  other,  how  many  have  you  in  both  ? 

8.  James  has  two  apples,  and  William  has  three ; 
if  James  gives  his  apples  to  William,  how  many  will 
William  have? 

9.  If  you  count  all  the  fingers   on  one  hand,  and 
two  on  the  other,  how  many  will  there  be  ? 

10.  George  has  three  cents,  and  Joseph  has  four; 
liow  many  have  they  both  together  ? 

*  The  first  questions  in  this  section  are  intended  for  very  young 
children.  It  will  be  well  for  the  instructor  to  giv  i  a  great  many 
more  of  this  kind.  Older  pupils  may  omit  these. 


20  AKITHMETIC.  [Part  1 

11.  Robert  gave  five  cents  for  an  orange,  and  two 
for  an  apple ;  how  many  did  he  give  for  both  ? 

12.  If  a  custard  cost  six  cents,  and  an  apple  two 
cents,  how  many  cents  will  it  take  to  buy  an  apple  and 
a  custard  ? 

13.  If  you  buy  a  pint  of  nuts  for  five  cents,  and  an 
orange  for  three  cents,  how  many  cents  would  you 
give  for  both  ?  hu\v  many  more  for  the  nuts  than  for 
the  orange? 

14.  If  an  ounce  of  figs  is  worth  six  cents,  and  a 
half  a  pint  of  cherries  is  worth  three  cents,  how  much 
are  they  both  worth  ? 

15.  Dick  had  five  plums,  and  John  gave  him  four 
more;  how  many  had  he  then? 

1 6.  How  many  fingers  have  you  on  both  hands  ? 

17.  How  many  fingers  and   thumbs  have  you  on 
both  hands? 

18.  If  you  had  six  marbles  in  one  hand,  and  four 
in  the  other,  how  many  would  you  have  in  the  one 
more  than  in  the  other?  how  many  would  you  have 
in  both  hands  ? 

19.  David  had  seven  nuts,  and  gave  three  of  them 
to  George;  how  many  had  he  left? 

20.  Two  boys,  James  and    Robert,  bought  seven 
marbles  apiece  ;  James  gave  Robert  four  of  Ids  ;  how 
many  had  each  then  ? 

21.  A  boy,  having  eleven  nuts,  gave  away  three  of 
them ;  how  many  had  he  left  ? 

22.  If  you  had  eight  cents,  and  your  papa  should 
give  ypu  five  more,  how  many  would  you  have  ? 

23.  A  man  bought  a  sheep  for  eight  dollars,  and 
a    calf   for    seven   dollars ;  what    did    he    give    for 
both? 

24.  A  man  bought  a  barrel  of  flour  for  eight  dol- 
lars, and  sold  it  for  four  dollars  more  than  he  gave  foi 
it ;  how  much  did  he  sell  it  for  ? 

25.  A  man  bought  a  hundred-weight  of  su^ar  foi 


Sect.  I.]  ARITHMETIC  21 

nine  dollars,  and  a  barrel  of  flour  for  seven  dollars 
how  much  did  he  give  for  the  whole  ? 

26.  A  man  bought  three  barrels  of  cider  for  eight 
dollars,  and  ten  bushels  of  apples  for  nine  dollars 
how  much  did  he  give  for  the  whole  ? 

27.  A  man  bought  a  firkin  of  butter  for  twelve  dol- 
lars, but,  it  being  damaged,  he  sold  it  again  for  eight 
dollars  ;  how  much  did  he  lose  ?    • 

28.  A  man  bought  three  sheep  for  fifteen  dollars, 
but  could  not  sell  them  again  for  so  much  by  eigh* 
dollars  ;  how  much  did  he  sell  them  for  ? 

29.  A  man  bought  sixteen  pounds  of  coffee,  and  lost 
seven  pounds  of  it  as  he  was  carrying  it  home  ;  how 
much  had  he  left? 

30.  A  man  bought  nineteen  pounds  of  sugar,  and 
having  lost  a  part  of  it,  he  found  he  had  nine  pounds 
left ;  how  much  had  he  lost  ? 

31.  A  man,  owing  fifteen  dollars,  paid  nine  dollars 
of  it ;  how  much  did  he  then  owe  ? 

32.  A  man,  owing  seventeen  dollars,  paid  all  but 
seven  dollars ;  how  much  did  he  pay  ? 

B.  1.  Two  and  one  are  how  many  ? 

2.  Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 

4.  Four  and  two  are  how  many  ? 

5.  Five  and  two  are  how  many  ? 

6.  Six  and  two  are  how  many  ? 

7.  Seven  and  two  are  how  many  ? 

8.  Eight  and  two  are  how  many  ? 

9.  Nine  and  two  are  how  many  ? 

10.  Ten  and  two  are  how  many  ? 

1 1 .  Two  and  three  are  how  many  ? 

12.  Three  and  three  are  how  many  ? 

1 3.  Four  and  three  are  how  many  ? 

14.  Five  and  three  are  how  many  ? 

15.  Six  an/,  three  are  how  many  ? 


22  ARITHMETIC.  i  Azrf  I, 

16.  Seven  and  three  are  how  many? 

17.  Kijrht  and  three  are  how  many? 

18.  Nine  and  three  are  how  many? 

19.  Ten  and  three  are  how  many? 

20.  Two  and  lour  are  how  many? 

21.  Three  and  four  are  how  many  ? 

22.  Four  and  four  are  how  many? 

23.  Five  and  four  are  how  many!? 

24.  Six  and  four  are  how  many  ? 

25.  Seven  and  four  are  how  many? 

26.  Eight  and  four  are  how  many  ? 

27.  Nine  and  four  are  how  many  ? 

28.  Ten  and  four  are  how  many  ? 

29.  Two  ajid  five  are  how  many  ? 

30.  Three  and  five  are  how  many  ? 

31.  Four  and  five  are  how  many? 

32.  Five  and  five  are  how  many  ? 

33.  Six  and  live  are  how  many? 

34.  Seven  and  five  are  how  many  ? 

35.  Ei«jht  and  live  are  how  many  ? 

36.  Nine  and  five  are  how  many? 

37.  Ten  and  five  are  how  many  ? 

38.  Two  and  six  are  how  many  ? 

39.  Three  and  six  are  how  many  ? 

40.  Four  and  six  are  how  many  ? 

41.  Five  and  six  are  how  many  ? 

42.  Six  and  six  are  how  many  ? 

43.  Seven  and  six  are  how  many  ? 

44.  Eight  and  six  are  how  many  ? 

45.  Nine  and  six  are  how  many  ? 

46.  Ten  and  six  are  how  many  ? 

47.  Two  and  seven  are  how  many  ? 
48    Three  and  seven  are  how  many  ? 

49.  Four  and  seven  are  how  many  ? 

50.  Five  and  seven  are  how  many  ? 

51.  Six  and  seven  are  how  many  ? 

52.  Seven  and  seven  are  how  many  ? 


Seat.  1.1  ARITHMETIC.  23 

53.  Eight  and  seven  are  how  many  ? 

54.  Nine  and  seven  are  how  many  ? 

55.  Ten  and  seven  are  how  many  ? 

56.  Two  and  eight  are  how  many  ? 
57    Three  and  eight  are  how  many  ? 

58.  Four  and  eight  are  how  many  ? 

59.  Five  and  eight  are  how  many  ? 

60.  Six  and  eight  are  how  many  ? 

61.  Seven  and  eight  are  how  many  ? 

62.  Eight  and  eight  are  how  many  ? 

63.  Nine  and  eight  are  how  many  ? 

64.  Ten  and  eight  are  how  many  ? 

65.  Two  and  nine  are  how  many  ? 

66.  Three  and  nine  are  how  many  ? 

67.  Four  and  nine  are  how  many  ? 

68.  Five  and  nine  are  how  many  ? 

69.  Six  and  nine  are  how  many  ? 

70.  Seven  and  nine  are  how  many  ? 

71.  Eight  and  nine- are  how  many  ? 

72.  Nine  and  nine  are  how  many  ? 

73.  Ten  and  nine  are  how  many  ? 

74.  Two  and  ten  are  how  many  ? 

75.  Three  and  ten  are  how  many  ? 

76.  Four  and  ten  are  how  many  ? 

77.  Five  and  ten  are  how  many? 

78.  Six  and  ten  are  how  many  ? 

79.  Seven  and  ten  are  how  many  ? 

80.  Eight  and  ten  are  how  many  ? 

81.  Nine  and  ten  are  how  many  ? 

82.  Ten  and  ten  are  how  many  ? 

C.    1    Two  and  one  are  how  many  ? 
2    Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 

4.  Five  and  two  are  how  many  ? 

5.  Four  and  two  are  how  many  ? 

6.  Six  and  two  are  how  many  ? 


B4  ARITHMETIC.  [Part  1 

7.  Eight  and  two  are  how  many  ? 

8.  Five  and  three  are  how  many  ? 

9.  Seven  and  three  are  how  many? 

10.  Four  ami  three  are  how  many? 

1 1.  Two  and  three  are  how  many  ? 

12.  Two  and  six  are  how  many  ? 

13.  Two  and  eight  are  how  many  ? 

14.  Six  and  three  are  how  many? 

15.  Three  and  four  are  how  many  ? 

16.  Three  and  six  arc  how  many  ? 

1 7.  Two  and  seven  are  how  many  ? 

18.  Ten  and  two  are  how  many? 

19.  Two  and  four  are  how  many  ? 

20.  Three  and  seven  are  how  many  ? 

21.  Four  and  four  are  how  many  ? 

22.  Five  and  four  are  how  many  ? 

23.  Seven  and  two  are  how  many  ? 

24.  Two  and  five  are  how  many  ? 

25.  Three  and  three  are  how  many? 

26.  Four  and  five  are  how  many  ? 

27.  Nine  and  two  are  how  many  ? 

28.  Three  and  five  are  how  many  ? 

29.  Two  and  ten  are  how  many  ? 

30.  Three  and  eight  are  how  many  ? 

31.  Ten  and  three  are  how  many  ? 

32.  Two  and  nine  are  how  many  ? 

33.  Foui  and  six  are  how  many  ? 

34.  Eight  and  three  are  how  many  ? 

35.  Seven  and  four  are  how  many  ? 

36.  Nine  and  three  are  how  many  ? 

37.  Six  and  four  are  how  many  ? 

38.  Five  and  five  are  how  many  ? 

39.  Three  and  nine  are  how  many  ? 

40.  Four  and  seven  are  how  many  ? 

41.  Six  and  five  are  how  many  ? 

42.  Three  and  ten  are  how  many  ? 

43.  Eight  and  four  are  how  many  ? 


Sect.  1.]  ARITHMETIC.  25 

44.  Five  and  eight  are  how  many  ? 

45.  Four  and  nine  are  how  many  ? 

46.  Five  and  six  are  how  many  ? 

47.  Ten  and  four  are  how  many  ? 

48.  Seven  and  five  are  how  many  ?^ 

49.  Six  and  six  are  how  many  ? 

50.  Nine  and  four  are  how  many  ? 

51.  Eight  and  five  are  how  many  ? 

52.  Five  and  nine  are  how  many  ? 

53.  Four  and  ten  are  how  many  ? 

54.  Six  and  seven  are  how  many  ? 

55.  Four  and  eight  are  how  many  ? 

56.  Nine  and  five  are  how  many  ? 

57.  Six  and  eight  are  how  many  ? 

58.  Ten  and  five  are  how  many  ? 

59.  Seven  and  six  are  how  many  ? 

60.  Eight  and  seven  are  how  many  ? 

61.  Six  and  nine  are  how  many  ? 

62.  Seven  and  seven  are  how  many? 

63.  Eight  and  six  are  how  many  ? 

64.  Ten  and  six  are  how  many  ? 

65.  Eight  and  eight  are  how  many  ? 

66.  Nine  and  seven  are  how  many  ? 

67.  Ten  and  eight  are  how  many  ? 

68.  Six  and  ten  are  how  many  ? 

69.  Five  and  seven  are  how  many  ? 

70.  Nine  and  six  are  how  many  ? 

71.  Seven  and  eight  are  how  many  ? 

72.  Eight  and  nine  are  how  many  ? 

73.  Nine  and  nine  are  how  many  ? 

74.  Five  and  ten  are  how  many  ? 

75.  Seven  and  nine  are  how  many  ? 

76.  Nine  and  eight  are  how  many  ? 

77.  Eight  and  ten  are  how  many  ? 

78.  Ten  and  nine  are  jiow  many  ? 

79.  Seven  and  ten  .are  how  many? 

80.  Nine  and  ten  are  how  many  ? 

81.  Ten  and  ten  are  how  many  ? 


86  ARITHMETIC.  [Par/  1 

D.  1.  Three  boys,  Peter,  John,  and  Oliver,  gave 
some  money  to  a  beggar.  Peter  gave  seven  cents : 
John,  four  cents  ;  and  Oliver,  three  cents  ;  how  many 
did  they  all  give  him  ? 

2.  How  maiiy  did  Peter  give  more  than  Oliver  ? 

3.  Frank  had  nine  pears,  and  gave  three  of  them 
to   Harry  ;  how  many  had   he   left ;    and  how  many 
more  than  Harry  had  he  then? 

4.  Dick  had  ten  peaches,  Harry  twelve,  and  Charles 
thirteen;  Dick  gave  three  to  Stephen,   Harry  gave 
him  six,  and  Charles  gave  him  five;  how  many  had 
Stephen  ?  and  how  many  had  each  lefl  ? 

5.  A  boy  had  twenty  apples,  and  gave  them  to  his 
companions,  as  follows :  to  one  he  gave  three  ;  to  an- 
other, two  ;  to  another,   four ;  and  to  another,  five  ; 
how  many  did  he  give  away  ?  and  how  many  had  he 
left? 

6.  A  boy  gave  to   one  of   his   companions   eight 
peaches ;  to  another,  six ;  to  another,  four ;  and  kept 
two  himself;  how  many  had  he  at  first  ? 

7.  A  boy  went  to  the  confectioner's  and  bought 
three  cakes  of  gingerbread,  for  which  he  gave  a  cent 
apiece ;    two    buns,  for  which    he  gave   three  cents 
apiece ;  one  custard  for  four  cents,  and  one  orange  for 
six    cents ;   how  many  cents    did  he   spend    for   the 
whole  ? 

8.  A   boy,  having   twenty-five    cents,  bought   one 
quart  of  cherries  for  eight  cents,  one  orange  for  six 
cents,  and  gave  away  three  cents ;  how  many  cents 
had  he  left*? 

9.  A  boy,  bought  a  box  for  eighteen  cents,  and  gave 
eight  cents  to  have  it  painted,  and  then  sold  it  for 
thirty-two  cents  ;  how  much  did  he  gain  by  the  bar- 
gain ? 

10.  A  man  bought  a  sleigh  for  seventeen  dollars, 
and  gave  nine  dollars  to  have  it,  repaired  and  painted, 
and  then  sold  it  for  twenty-three  dollars  ;  how  much 
did  he  lose  by  the  bargain  ? 


Sect.  1.]  ARITHMETIC.  27 

11.  Eleven  and  two  are  how  many  ? 

1 2.  Eleven  and  three  are  how  many  ? 

1 3.  Eleven  and  four  are  how  many  ? 

1 4.  Eleven  and  five  are  how  many  ? 

1 5.  Eleven  and  six  are  how  many  ? 

1 6.  Eleven  and  seven  are  how  many  ? 

17.  Eleven  and  eight  are  how  many  ? 

18.  Eleven  and  nine  are  how  many? 

19.  Eleven  and  ten  are  how  many? 

20.  Twelve  and  two  are  how  many  ? 

21.  Twelve  and  three  are  how  many  ? 

22.  Twelve  and  four  are  how  many  ? 

23.  Twelve  and  five  are  how  many  ? 

24.  Twelve  and  six  are  how  many  ? 

25.  Twelve  and  seven  are  how  many  ? 

26.  Twelve  and  eight  are  how  many  ? 

27.  Twelve  and  nine  are  how  many  ? 

28.  Twelve  and  ten  are  how  many  ? 

29.  Thirteen  and  two  are  how  many? 

30.  Thirteen  and  three  are  how  many  ? 

31.  Thirteen  and  four  are  how  many  ? 

32.  Thirteen  and  five  are  how  many  ? 

33.  Thirteen  and  six  are  how  many  ? 

34.  Thirteen  and  seven  are  how  many  ? 

35.  Fourteen  and  two  are  how  many  ? 

36.  Fourteen  and  three  are  how  many  ? 

37.  Fourteen  and  four  are  how  many  ?  • 

38.  Fourteen  and  five  are  how  many  ? 

39.  Fourteen  and  six  are  how  many  ? 

40.  Fifteen  and  two  are  how  many  ? 

41.  Fifteen  and  three  are  how  many  ? 

42.  Fifteen  and  four  are  how  many  ? 

43.  Fifteen  and  five  are  hew  many? 

44.  Sixteen  and  two  are  how  many  ? 

45.  Sixteen  and  three  are  how  many  ? 

46.  Sixteen  and  four  are  how  many  ? 

47.  Seventeen  and  two  are  how  many  ? 


28  ARITHMETIC.  [Part  1. 

48.  Seventeen  and  three  are  how  many  ? 

49.  Eighteen  and  two  are  how  many? 

E.  1.  A  man  bo  ight  fx  sheep  for  nine  dollars,  and 
to  pay  for  it  he  gave  live  bushels  of  corn  worth  four 
dollars,  and  the  rest  in  money  ;  how  much  money  did 
he  pay  ? 

2.  If  a  barrel  of  flour  is  worth  eight  dollars,  and  a 
hundred-Weight  of  sugar  is  worth  twelve  dollars;  how 
much  more  is  the  sugar  worth  than  the  Hour  ? 

3.  If  a  man  had  eleven  dollars,  and  should  buy 
three  bushels  of  corn  for  five  dollars,  how  much  money 
would  he  have  left  ? 

4.  A  man  bought  a  firkin  of  butter  for  fifteen  dol- 
lars, but  it  being  damaged,  lie  was  willing   to  sell  it 
again  for  eight  dollars  less  than  he  gave  for  it;  what 
did  he  sell  it  for  ? 

5.  A  man  bought  three  barrels  of  flour  for  eighteen 
dollars,  and  sold  it  again  for  eleven  dollars  ;  what  did 
he  lose  by  the  bargain  ? 

6.  A  man  bought  a  keg  of  tobacco  for  thirteen  dol- 
lars,  and  sold  it  again  for  eighteen ;  what  did  he  gain 
by  the  bargain  ? 

7.  Five  less  two  are  how  many  ? 

8.  Seven  less  three  are  how  many  ? 

9.  Three  less  three  are  how  many  ? 

1 0.  Nine  less  three  are  how  many  ? 

11.  Six  less  two  are  how  many  ? 

12.  Seven  less  four  are  how  many  ? 

13.  Eight  less  three  are  how  many  ? 

14.  Five  less  four  are  how  many  ? 

1 5.  Seven  less  five  are  how  many  ? 

16.  Nine  less  five  are  how  many  ? 

17.  Eight  less  six  are  how  many  ? 

1 8.  Eleven  less  two  are  how  many  ? 

19.  Twelve  less  four  are  how  many? 


Sect.  1]  ARITHMETIC.  M 

20.  Ten  less  seven  are  how  many  ? 

21.  Thirteen  less  five  are  how  many  ? 

22.  Fourteen  less  eight  are  how  many  ? 

23.  Twelve  less  seven  are  how  many  ? 

24.  Seventeen  less  five  are  how  many  ? 

25.  Eighteen  less  ten  are  how  many  ? 
^26.  Thirteen  less  seven  are  how  many  ? 

27.  Sixteen  less  seven  are  how  many  ? 

28.  Fifteen  less  seven  are  how  many  ? 

29.  Nineteen  less  six  are  how  many  ? 

30.  Eighteen  less  five  are  how  many  ? 

31.  Seventeen  less  eight  are  how  many  ? 

32.  Fourteen  less  nine  are  how  many  ? 

33.  Sixteen  less  five  are  how  many  ? 

34.  Fifteen  less  eight  are  how  many  ? 

35.  Fourteen  less  nine  are  how  many  ? 

36.  Sixteen  less  ten  are  how  many? 

37.  Seventeen  less  nine  are  how  many  ? 

38.  Eighteen  less  seven  are  how  many  ? 

F.  1.  How  many  are  nine  and  two  ?  Nineteen  and 
two  ?  Twenty-nine  and  two  ?  Thirty-nine  and  two  ? 
Forty-nine  and  two  ?  Fifty-nine  and  two  ?  Sixty-nine 
and  two  ?  Seventy-nine  and  two  ?  Eighty-nine  arid 
two  ?  Ninety-nine  and  two  ? 

2.  How  many  are  nine  and  three?  Nineteen  and 
three  ?    Twenty-nine   and    three  ?    Thirty-nine    and 

hree  ?  Forty-nine  and  three  ?  Fifty-nine  •  and  three  ? 
Sixty-nine  and  three  ?  Seventy-nine  and  three  ? 
Eighty-nine  and  three?  Ninety-  .me  and  three  ? 

3.  How  many  are  nine  and  four  ?  Nineteen  and 
four?  Twenty-nine  and  four?    .thirty-nine  and  four? 
Forty-nine  and  four  ?    Fifty-nine  and  four  ?    Sixty- 
aine  and  four  ?  Seventy-nine  and  four  ?  Eighty-nine 
and  four  ?  Ninety-nine  and  four  ? 

4.  How  many  are  nine  and  five  ?   Nineteen  and 
Gve  ?  Twenty-nine  and  five  ?    Thirty- nine  and  five  ? 


3U  ARITHMETIC.  [Pan  t 

Forty-nine  and  five  ?  Fifty-nine  and  five  ?  Sixty-nine 
and  live  ?  Seventy-nine  and  five  ?  Eighty-nine  and 
five  ?  Ninety-nine  and  five  ? 

5.  How   many  are    nine    and    six?    Nineteen   and 
six  ?    Twenty-nine    and    six  ?    Thirty-nine   and    six : 
Forty-nine  and  six?   Fifty-nine  and  six '{  Sixty-nine 
and  six?  Seventy-nine  and  six?  Eighty-nine  and  six  ? 
Ninety-nine  and  six  ? 

6.  How  many  are  nine  and  seven  ?  Nineteen  and 
seven  ?    Twenty-nine   and   seven  ?    Thirty-nine   and 
seven  ?  Forty-nine  and  seven  ?  Fifty-nine  and  seven  ? 
Sixty-nine    and    seven  ?    Seventy-nine    and   seven  ? 
Eighty-nine  and  seven?     Ninety-nine  and  seven? 

7.  How  many  are  nine  and  eight  ?   Nineteen  and 
eight?    Twenty-nine    and     eight?    Thirty-nine    and 
eight?  Forty-nine  and  eight?   Fifty-nine  and  eight? 
Sixty-nine    and    eight?    .Seventy-nine     and     eight? 
Eighty-nine  and  eight  ?     Ninety-nine  and  eight  ? 

8.  How  many  are  nine  and   nine  ?  Nineteen  and 
nine  ?  Twenty-nine  and  nine  ?  Thirty-nine  and  nine  ? 
Forty-Bine  and  nine?  Fifty-nine  and  nine  ?  Sixty-nine 
and  nine  ?  Seventy-nine  and  nine?  Eighty-nine  and 
nine  ?  Ninety-nine  and  nine  ? 

9.  How  many  are  nine  and  ten  ?  Nineteen  and  ten? 
Twenty-nine  and  ten  ?  Thirty-nine  and  ten  ?  Forty- 
nine  and  ten  ?  Fifty-nine  and  ten  ?    Sixty-nine  and 
ten?  Seventy-nine   arid   ten?    Eighty-nine    and  ten? 
Ninety-nine  and  ten  ? 

10.  How  many  are    eight  and   three?     Eighteen 
and    tliree  ?    Tweiu  '-eight   and    three  ?    Tliirty-eight 
and  three?    Forty-e*  lit  and  three?    Fifty-eight  and 
tliree  ?     Sixty-eight  a,ad  three  ?     Seventy-eight   and 
three  ?    Eighty-eight   and   tliree  ?    Ninety-eight   and 
three  ? 

11.  How  many  are  eight  and  four?  Eighteen  and 
four  ?  Twenty-eight  and  four  ?  Thirty-eight  and  lour  ? 
vorty-eight  and   four?  Fifty-eight  and  four?  Sixty- 


Sect.  1.]  ARITHMETIC.  31 

eight  and  four  ?  Seventy-eight  and  four  ?  Eighty-eight 
and  four  ?  Ninety -eight  and  four  ? 

12.  How  many  are  eight  and  five?  Eighteen  and 
five  ?  Twenty-eight  and  five  ?  Thirty-eight  and  five  ? 
Forty-eight  and  five  ?    Fifty-eight  and  five  ?    Sixty- 
eight  and  five  ?  Seventy-eight  and  five  ?  Eighty-eight 
and  five  ?  Ninety-eight  and  five  ? 

13.  How  many  are  eight  and  six?  Eighteen  and 
six  ?  Twenty-eight  and  six  ?    Thirty-eight  and  six  ? 
Forty-eight  and  six  ?  Fifty-eight  and  six  ?  Sixty-eight 
and  six  ?  Seventy-eight  and   six  ?    Eighty-eight  and 
six?  Ninety-eight  and  six? 

14.  How  many   are   eight  and  seven  ?     Eighteen 
and  seven?    Twenty-eight  and  seven?    Thirty-eight 
and  seven  ?    Forty-eight  and  seven  ?    Fifty-eight  and 
seven  ?   Sixty-eight  and  seven  ?      Seventy-eight  and 
seven  ?     Eighty-eight  and  seven  ?     Ninety-eight  and 
seven  ? 

15.  How  many  are  eight  and  eight?    Eighteen  and 
eight  ?     Twenty-eight  and  eight  ?     Thirty-eight  and 
eight  ?  Forty-eight  and  eight  ?   Fifty-eight  and  eight  ? 
Sixty-eight  and   eight  ?      Seventy-eight  and  eight  ? 
Eighty-eight  and  eight  ?     Ninety -eight  and  eight  ? 

1 6.  How  many  are  eight  and  nine  ?     Eighteen  and 
nine  ?  Twenty-eight  and  nine  ?  Thirty-eight  and  nine  ? 
Forty-eight  and  nine  ?    Fifty-eight  arid  nine  ?    Sixty- 
eight  and  nine  ?     Seventy-eight  and  nine  ?     Eighty- 
eight  and  nine  ?     Ninety-eight  and*  nine  ? 

17.  How  many  are  seven  and  four  i     Seventeen 
and  four  ?     Twenty-seven  and  four  ?     Thirty-seven 
and  four  ?     Forty-seven  and  four  ?     Fifty-seven  and 
four  ?      Sixty-seven  and  four  ?      Seventy-seven  and 
four?      Eighty-seven  and  four?     Ninety-seven  and 
four  ? 

18.  How  many  are  seven  and  five  ?  Seventeen  and 
five  ?  Twenty-seven  and  five  ?  Thirty-seven  and  five 
Forty-seven  and  five  ?    Fifty-seven  and  five  ?    Sixty 


32  ARITHMETIC.  [Part  1 

seven  and  five?     Seventy-seven  and  five?     Eighty- 
seven  and  five  ?     Ninety-seven  and  five  ? 

19.  How  many  are  seven  and  six  ?  Seventeen  and 
six?    Twenty-seven  and  six?    Thirty-seven  and  six? 
Forty-seven  and  six  ?     Fifty-seven  and  six  ?     Sixty- 
seven  and  six?  Seventy-seven  and  six?  Eighty-seven 
and  six  ?    Ninety-seven  and  six  ? 

20.  How  many  are  seven  and  seven  ?    Seventeen 
and  seven  ?    Twenty-seven  and  seven  ?    Thirty-seven 
and  seven  ?    Forty-seven  and  seven  ?   Fifty-seven  and 
seven  ?    Sixty-seven  and  seven  ?    Seventy-seven  and 
seven?    Eighty-seven  and  seven  ?    Ninety-seven  and 
seven  ? 

21.  How  many  are  seven  and  eight  ?     Seventeen 
and  eight?    Twenty-seven  and  eight?    Thirty-seven 
and  eight  ?    Forty-seven  and  eight  ?    Fifty-seven  and 
eight  ?    Sixty-seven  and  eight  ?    Seventy-seven  and 
eight?    Eighty-seven  and  eight?    Ninety-seven  and 
eight? 

'2'2.  How  many  are  seven  and  nine?  Seventeen  and 
nine  ?  Twenty-seven  and  nine  ?  Thirty-seven  and 
nine  ?  Forty-seven  and  nine  ?  Fifty -seven  and  nine  ? 
Sixty-seven  and  nine  ?  Seventy-seven  and  nine  ? 
Eighty-seven  and  nine  ?  Ninety-seven  and  nine  ? 

23.  How  many  are  six  and  five  ?  Sixteen  and  five  ? 
Twenty-six  and  five  ?  Thirty-six  and  five  ?   Forty-six 
and  five  ?     Fifty-six  and  five  ?     Sixty-six  and  five  ? 
Seventy-six  and  fiVe  ?    Eighty-six  and  five  ?    Ninety- 
six  and  five  ? 

24.  How  many  are  six  and  six  ?    Sixteen  and  six  ? 
Twenty-six  and  six  ?    Thirty-six  and  six  ?     Forty-six 
and  six  ?  Fifty-six  and  six  ?   Sixty-six  and  six  ?   Sev- 
enty-six and  six  ?    Eighty-six  and  six  ?    Ninety-six 
and  six  ? 

25.  How  many  are  six  and  seven?     Sixteen  and 
seven?  Twenty-six  and  seven  ?  Thirty-six  and  seven  ? 
Forty-six  and  seven  ?  Fifty-six  and  seven  ?  Sixty -six 


Sect,  i.j  ARITHMETIC.  33 

and  seven  ?    Seventy-six  and  seven  ?    Eighty-six  and 
seven  ?    Ninety-six  and  seven  ? 

26.  How  many  are  six  and  eight?     Sixteen  and 
eight  ?    Twenty -six  and  eight  ?    Thirty-six  and  eight  ? 
Forty-six  and  eight  ?    Fifty-six  and  eight  ?    Sixty-six 
and  eight  ?    Seventy-six  and  eight  ?    Eighty-six  and 
eight  ?    Ninety-six  and  eight  ? 

27.  How  many   are   six   and   nine?     Sixteen  and 
nine  ?    Twenty-six  and  nine  ?    Thirty-six  and  nine  ? 
Forty-six  and  mne  ?    Fifty-six  and  nine?     Sixty-six 
and   nine  ?     seventy-six   and  nine  ?     Eighty -six  and 
nine  ?    Ninety-six  and   nine  ? 

28.  How  many  are  five  and  six?    Fifteen  and  six? 
Twenty-five  and  six  ?    Thirty-five  and  six  ?    Forty- 
five  and  six  ?    Fifty-five  and  six  ?    Sixty-five  and  six  ? 
Seventy-five  and  six  ?    Eighty-five  and  six  ?    Ninety- 
five  and  six  ? 

29.  How  many  are  five  and  seven  ?     Fifteen  and 
seven  ?     Twenty-five    and    seven  ?     Thirty-five   and 
seven  ?    Forty-five  and  seven  ?    Fifty-five  and  seven? 
Sixty-five  and  seven  ?  Seventy-five  and  seven  ?  Eigh- 
ty-five and  seven  ?    Ninety-five  and  seven  ? 

30.  I  low  many  arc  five  and  eight  ?    Fifteen   and 
eight  ?      Twenty-five    and    eight  ?      Thirty-five   and 
eight?     Forty-five  and  eight?     Fifty-five  and  eight? 
Sixty-five  and  eight  ?    Seventy -five  and  eight?    Eigh- 
ty-five and  .eight  ?    Ninety-five  and  eight  ? 

31.  How   many  are  five   and  nine  ?     Fifteen  and 
nine?    Twenty-five  and  nine?    Thirty-five  and  nine? 
Pbrty-five  and  nine  ?    Fifty-five  and  nine  ?    Sixty-five 
and  nine  ?    Seventy-five  and  nine  ?    Eighty-five  and 
nine  ?    Ninety-five  and  nine  ? 

32.  How  many  are  four  and  seven  ?    Fourteen  and 
seven  ?     Twenty-four  and  seven  ?     Thirty-four  and 
seven  ?  Forty-four  and  seven  ?   Fifty-four  and  seven  ? 
Sixty-four   and   seven  ?      Seventy-four   and    seven  ? 
Eighty-four  and  seven  ?     Ninety-four  and  seven  ? 

3 


34  ARITHMETIC.  \Part  1 

33.  How  many  are  four  and  eight  ?     Fourteen  and 
eight  ?      Twenty-four  and  eight  ?      Thirty-four  and 
eight  ?     Forty-four  and  eight  ?     Fifty-four  and  i-ight  ? 
Sixty-four  and  eight  ?  Seventy-four  and  eight  ?   Eigh- 
ty-four and  eight  ?    Ninety-four  and  eight  ? 

34.  How  many  are  four  and  nine?    F'ourteen  and 
nine  ?    Twenty-four  and  nine  ?    Thirty-four  and  nine  ? 
Forty-four  and  nine  ?  Fifty-four  and  nine  ?  Sixty-four 
and  nine  ?    Seventy-four  and  nine  ?    Eighty-four  and 
nine  ?    Ninety-four  and  nine  ? 

35.  How  many  are   three   and   eight  ?     Thirteen 
and  eight?     Twenty-three  and  eight?     Thirty- tliree 
and  eight  ?     Forty-three  and  eight  ?    Fifty-three  and 
eight  ?     Sixty-three  and  eight  ?     Seventy-three  and 
fijrht  ?     Eighty-three  and  eight  ?     Ninety-three  and 
eight  ? 

36.  How  manj  are  three  and  nin^  ?    Thirteen  and 
dine  ?     Twenty-three   and   nine  ?      Thirty-three  and 
nine?    Forty-three  and  nine?    Fifty-three  and  nine  ? 
Sixty-three  and  nine  ?  Seventy -three  and  nine  ?  Eigh- 
ty-three and  nine  ?    Ninety-three  and  nine  ? 

37.  How  many  are  two  and  nine  ?     Twelve  and 
nine  ?    Twenty-two  and  nine  ?    Thirty-two  and  nine  ? 
Forty-two  and  nine  ?    Fifty-two  and  nine  ?    Sixty -two 
ami  nine  ?    Seventy-two  and  nine  ?    Eighty-two  and 
nine  ?    Ninety-two  and  nine  ? 

G.  1.  A  man  bought  a  firkin  of  butter  for  nine 
dollars,  a  keg  of  molasses  for  six  dollars,  and  five 
bushels  of  wheat  for  seven  dollars ;  how  much  did  he 
give  for  the  whole  ? 

2.  A  boy  gave  some  apples  to  his  companions :  to 
one  he  gave  seven,  to  another  six,  and  to  another 
liijrht ;  how  many  did  he  give  to  the  whole  ? 

3.  A  man  bought  a  cow  for  seventeen  dollars,  a 
sheep  for  nine,  and  a  calf  for  seven ;  how  much  did 
hi   give  for  the  whole  ? 


Sect.  1.]  ARITHMETIC.  35 

4.  A  drover  bought  sheep  as  follows:  of  one  man 
he  bought   twenty-seven,  of  another  eight,  of  another 
ten,  HIH!  of  another  five  ;  afterwards  he  sold  nine  of 
them  ;    how  many   had  he  then  ? 

5.  A  lady  bought  a   comb   for   thirty-seven  cents> 
some   tape  for   eight    cents,  some   pins  for  ten   cents, 
some  needles  for  six  cents,  and  some  thread  for  six 
cents ;  she  gave  seventy  -five  cents  ;  how  much  change 
ought  she  to  receive  back? 

6.  Eight,  and  nine,  and  six,  are  how  many? 

7.  Five,  and  seven,  and  three,  are  how  many  ? 

8.  Four,  and  three,  less  two,  are  how  many  ? 

9.  Seven,  and  five,  less  three,  are  how  many  ? 

10.  Sixteen,  and  nine,  and  three,  are  how  many  ? 

11.  Twenty-three  and  eight  are  how  many  ? 

12.  Twenty-seven  and  five  are  how  many  ? 

13.  Twenty-five,  less  eight,  are  how  many  ? 

14.  Thirty-two  and  seven,  less  nine,  are  how  many  ? 

15.  Thirty-eight,  and  six,  and  four,  less  seven,  are 
how  many  ? 

1 6.  Forty-four,  and  eight,  and  three,  and  seven,  are 
how  many? 

17.  Fifty-two,   and   six,   and   four,   and    five,  and 
three,  are  how  many  ? 

18.  Fifty-seven,  and  six,  and  three,  and  five,  and 
two,  less  eight,  are  how  many? 

19.  Sixty-three,   and  five,  and   four,  and  six,  and 
two,  less  seven,  are  how  many  ? 

20.  Seventy-five,  and  six,  and  eight,  and  three,  and 
seven,  and  four,  less  nine,  are  how  many  ? 

21.  Eighty-three,  and  six,  and  five,  and  two,  and 
*even,  and  nine,  less  four,  are  how  many  ? 

22.  Fifty-eight,  and   ten,  and  five,  and  seven,  and 
three,  and  six,  and  four,  less  nine,  are  how  many  ? 

23.  Sixty-seven,  and  five,  and  eight,  and  nine,  and 
seven,  less  six,  are  how  many  ? 

24.  Severity-four,  and  nine,  arid  seven,  and  five,  and 
two,  less  six,  are  how  many  ? 


36  ARITHMETIC.  [Part  L 

25.  Seventy-eight,  and  seven,  and  six,  and  two,  and 
five,  and  eight,  less  nine,  are  how  many  ? 

26.  Eighty-four,  and  seven,  and  six,  and  eight,  and 
five,  less  ton,  are  how  many? 

27.  Forty-seven,  and  eight,  and  six,  and  two,  and 
four,  and  eiirht,  and   three,  and  seven,  and  ten,  and 
nine,  less   five,  are  how   many? 

28.  Thirty-five,  and  eight,  and  four,  and  six,  and 
three,  and  four,  less  eleven,  are  how  many  ? 

29.  Seventy,  and  ten,  and  six,  and  nine,  and  seven, 
and  two,  and  five,  and  eight,  and  nine,  less  three,  are 
how  many  ? 

H.  1.  A  man  bought  a  cow  for  twenty-eight  dol- 
lars, and  a  sheep  for  four  dollars,  and  a  pig  for  seven 
dollars  ;  how  much  did  he  give  for  the  whole  ? 

2.  James  had  twenty-seven  cents  ;  John  gave  him 
four  more,  David  seven,  and  George  eleven,  and  he 
bought  nine  cents'  worth  of  cake ;   how  many  cents 
had  he  left? 

3.  A  man  paid  sixteen  dollars  to  A,  nine  dollars  to 
B,  seven  dollars  to  C,  ten  dollars  to  D,  six  dollars  to 
E,  four  dollars  to  F,  and  had  eight  dollars  left ;  how 
many  had  he  at  first  ? 

4.  From  Boston  to  Roxbury  it  is  three  miles  ;  from 
Roxbury  to    Dedham,  six  miles ;   from   Dedham  to 
Walpole,.  eleven  miles  ;  from  Walpole  to  Wrentham, 
four    miles ;   from   Wrentham    to   Attlehorough,  four 
miles ;  from  Attleborough  to  Pawtucket,  nine  miles ; 
from  Pawtucket  to  Providence,  four  miles ;  how  many 
miles  is  it  from  Boston  to  Providence  ? 

5.  One  boy  had  fifteen  nuts  ;  another  boy  gave  him 
seven  ;  another,  nine  ;  and  another  gave  him  enough 
to  make  his  number  forty ;  how  many  did  the  last  boy 
give  him  ? 

6.  A  boy  had  thirty-seven  apples ;  he  gave  five  to 
one  companion ;  and  eight  to  another ;  and  when  he 


Sect.  2.]  ARITHMETIC.  37 

had  given  some  to  another,  he  had  six  left ;  how  many 
did  lie  give  to  the  last  ? 

7.  A  man  owed  fifty-six  dollars  ;  at  one  time  h« 
paid  seventeen  dollars ;  at  another,  eight ;  at  another, 
five ;  at  another,  seven  ;  at  last  he  paid  the  rest  of  the 
debt,  wanting  four  dollars  ;   how  much  was  the  last 
payment  ? 

8.  Six  men  bought  a  horse  for  seventy  dollars  ;  the 
first  gave  twenty-three  dollars ;   the  second,  fifteen , 
the  third,  twelve  ;  the  fourth,  nine ;  the  fifth,  se^en ; 
how  much  did  the  sixth  give  ? 

9.  A  man  bought  a  horse  for  forty-five  dollars  and 
paid  fifteen  dollars  for  keeping  him ;  he  let  him  enough 
to  receive  twenty  dollars  ;  and  then  sold  him  for  forty- 
three  dollars ;  did  he  gain  or  lose  by  the  bargain  ?  and 
how  much  ? 


SECTION  H. 

A.  1.  WHAT  cost  three  yards  of  tape,  at  two  cents 
a  yard  ?  * 

2.  What  cost  four  apples,  at  two  cents  apiece  ? 

3.  What  cost  five  peaches,  at  three  cents  apiece  ? 

4.  What  must  you  give  for  two  oranges,  at  six  cents 
apiece  ? 

5.  What  would  be  the  price  of  three  barrels  of 
cider,  at  three  dollars  a  barrel  ? 

6.  If  one  orange  is  worth  three  apples,  how  many 
apples  are  four  oranges  worth  ? 

7.  What   are  two    barrels  of  flour  worth,  at  five 
dollars  a  barrel  ? 

*  The  pupil  should  be  made  to  observe  that  three  yards  will  cost 
three  times  as  much  as  one  yard ;  and  say,  if  one  yard  cost  two 
cents,  three  yards  will  cost  three  times  two  cents.  "He  should  be 
made  to  give  this  reason  for  the  solution  of  each  question,  varying 
the  numbers  according  to  the  question. 


88  ARITHMETIC/  [Part  1 

8.  What  cost  three  yards  of  cloth,  at  four  dollars  a 
yara  ? 

9.  What  cost  two  pounds  of  raisins,  at  eight  centa 
a  pound  ? 

10.  What  cost  three  lemons,  at  six  cents  apiece  ? 

11.  If  a  man  travel  three  miles  in  an  hour,  how 
many  miles  will  he  travel  in  four  hours? 

12.  What  will  live  pair  of  shoes  come  to  at  two 
dollars  a  pair  ? 

13.  What  is  the  price  of  seven  yards  of  cloth,  at 
three  dollars  a  yard  ? 

14.  What  is  the  value  of  two  pounds  of  beef,  at 
seven  cents  a  pound  ? 

15.  If  there  are  three  feet  in  one  yard,  how  many 
feet  are  there  in  four  yards  ? 

1 6.  How  many  feet  are  there  in  seven  yards  ? 

17.  How  many  feet  are  there  in  six  yards  and  two 
feet? 

18.  If  a  man  earn  seven  dollars  in  one  week,  how 
much  would  he  earn  in  five  weeks? 

19.  What  cost  seven  hundred-weight  of  sugar,  at 
nine  dollars  a  hundred-weight  ? 

20.  What  cost  seven  pounds  of  sugar,  at  ten  centa 
a  pound  ? 

21.  If  one   half  yard   of  cloth  cost   three  dollars, 
what  would  three  yards  cost  ? 

22.  If  one  quarter  of  a  yard  of  cloth  cost  two  dol- 
lars, what  is  that  a  yard  ? 

23.  How. many  yards  of  cloth  are  there  in  seven 
pieces,  each  piece  containing  ten  yards  ? 

24.  What  will  live  barrels  of  flour  cost,  at  six  dol- 
lars a  barrel  ? 

25.  If  a  man  can  travel  four  miles  in  an  hour,  how 
far  can  he  travel  in  eight  hours  ? 

26.  If  it  take  four  bushels  of  wheat  to  make  a  bar- 
rel of  flour,  how  many  bushels  will  it  take  to  make 
seven  barrels  ? 


Sect.  2.]  ARITHMETIC. 

B.  1.  Two  times  one  are  how  many  ?  * 

2.  Two  time's  two  are  how  many  ? 

3.  Two  times  three  are  how  many  ? 

4.  Two  times  four  are  how  many  ? 

5.  Two  times  five  are  how  many  ? 

6.  Two  times  six  are  how  many  ? 

7.  Two  times  seven  are  how  many 

8.  Two  times  eight  are  how  many  ? 
9    Two  times  nine  are  how  many  ? 

1 0    Two  time?  ten  are  how  many  ? 

1 1.  Three  times  one  are  how  many  ? 

12.  Three  times  two  are  how  many  ? 

13.  Three  times  three  are  how  many  ? 

14.  Three  times  four  are  how  many  ? 

15.  Three  tunes  five  are  how  many  ? 

1 6.  Three  times  six  are  how  many  ? 

17.  Three  times  seven  are  how  many? 

18.  Three  times  eight  are  how  many  r 
Itf.  Three  times  nine  are  how  many  ? 

20.  Three  times  ten  are  how  many  ? 

21.  Four  times  one  are  how  many  ? 

22.  Four  times  two  are  how  many  ? 

23.  Four  times  three  are  how  many  ? 

24.  F*our  times  four  are  how  many  ? 

25.  Four  times  five  are  how  many  ? 

26.  Four  times  six  are  how  many  ? 

27.  Four  times  seven  are  how  many? 

28.  Four  times  eight  are  how  many  ? 

29.  Four  times  nine  are  how  many  ? 

30.  Four  times  ten  are  how  many  ? 

31.  Five  times  one  are  how  many  ? 

32.  Five  times  two  are  how  many  ? 
83.  Five  times  three  are  how  many  ? 

34.  F^ive  times  four  are  how  many  ? 

35.  Five  times  five  are  how  many  ? 

36.  Five  times  six  are  how  many  ? 

*  See  the  Key. 


40  ARITHMETIC.  (Pan  I 

37.  Five  times  seven  are  how  many  ? 

38.  Five  times  eight  are  how  mariy  ? 

39.  Five  times  nine  are  how  many  ? 

40.  Five  times  ten  are  how  many  ? 

41.  Six  times  one  are  how  many  ? 

42.  Six  times  two  are  how  many  ? 

43.  Six  times  three  are  how  many  ? 

44.  Six  times  four  are  how  many  ? 

45.  Six  times  five  are  how  many  ? 

46.  Six  times  six  are  how  many  ? 

47.  Six  tunes  seven  are  how  many  ? 

48.  Six  times  eight  are  how  many  ? 

49.  Six  times  nine  are  how  many  ? 

50.  Six  times  ten  are  how  many  ? 

51.  Seven  times  one  are  how  many  ? 

52.  Seven  times  two  are  how  many  ? 

53.  Seven  times  three  are  how  many  ? 

54.  Seven  times  four  are  how  many  ? 

55.  Seven  times  five  are  how  many  ? 

56.  Seven  times  six  are  how  many  ? 

57.  Seven  times  seven  are  how  many  ? 

58.  Seven  times  eight  are  how  many  ? 

59.  Seven  times  nine  are  how  many  ? 

60.  Seven  tunes  ten  are  how  many  ? 

61.  Eight  times  one  are  how  many  ? 

62.  Eight  tunes  two  are  how  many  ? 

63.  Eight  times  three  are  how  many  ? 

64.  Eight  times  *bur  are  how  many  ? 

65.  Eight  times  five  are  how  many  ? 

66.  Eight  times  six  are  how  many  ? 

67.  Eight  times  seven  are  how  many  ? 

68.  Eight  times  eight  are  how  many  ? 

69.  Eight  times  nine  are  how  many  ? 

70.  Eight  times  ten  are  how  many  ? 

71.  Nine  times  one  are  how  many  ? 

72.  Nine  times  two  are  how  many  ? 

73.  Nine  times  three  are  how  many  ? 


Sect.  8.]  ARITHMETIC.  41 

74.  Nine  times  four  are  how  many  ? 

75.  Nine  times  five  are  how  many  ? 

76.  Nine  times  six  are  how  many  ? 

77.  Nine  times  seven  are  how  many  ? 

78.  Nine  times  eight  are  how  many  ? 

79.  Nine  times  nine  are  how  many  ? 

80.  Nine  times  ten  are  how  many  ? 

81.  Ten  times  one  are  how  many  ? 

82.  Ten  times  two  are  how  many  ? 

83.  Ten  times  three  are  how  many  ? 

84.  Ten  times  four  are  how  many  ? 

85.  Ten  times  five  are  how  many  ? 

86.  Ten  times  six  are  how  many  ? 

87.  Ten  times  seven  are  how  many  ? 

88.  Ten  times  eight  are  how  many  ? 

89.  Ten  times  nine  are  how  many  ? 

90.  Ten  times  ten  are  how  many  ? 

C.  1.  Two  times  two  are  how  many  times  one  ? 

2.  Three  times  two  are  how  many  times  one 

3.  Four  times  two  are  how  many  times  one  ? 

4.  Five  times  two  are  how  many  times  one  ? 

5.  Seven  times  two  are  how  many  ? 

6.  Nine  times  two  are  how  many  ? 

7.  Six  times  two  are  how  many  ? 

8.  Eight  times  two  are  how  many  ? 

9.  Ten  times  two  are  how  many  ? 

10.  Two  times  three  are  how  many  ? 

11.  Three  times  three  are  how  many  ? 

12.  Four  times  three  are  how  many  ? 
13. .  Five  times  three  are  how  many  ? 

14.  Six  times  three  are  how  many  ? 

15.  Eight  times  three  are  how  many  ? 

16.  Seven  times  three  are  how  many  ? 

17.  Ten  times  three  are  how  many  ?  • 

18.  Nine  times  three  are  how  many  ? 

19.  Two  times  four  are  how  many  ? 


42  ARITHMETIC. 

2u.  Six  times  four  are  how  many  ? 

21.  Four  times  four  are  how  many  ? 

22.  Seven  times  four  are  how  many  ? 

23.  Nine  times  four  are  how  many  ? 

24.  Three  times  four  are  how  many  ? 
2o.    Five  times  four  are  how  many  ? 

26.  Ten  times  four  are  how  many  ? 

27.  Eight  times  four  are  how  many? 

28.  Two  times  five  are  how  many  ? 

29.  Five  times  live  are  how  many  ? 

30.  Three  times  live  are  how  many  ? 

31.  Six  times  live  are  how  many? 

32.  Two  times  six  are  how  many? 

33.  Four  times  five  are  how  many  ? 

34.  Seven  times  five  are  how  many  ? 

35.  Three  times  six  are  how  many  ? 

36.  Seven  times  six  are  how  many  ? 

37.  Seven  tunes  seven  are  how  many  ? 

38.  Four  times  eight  are  how  many  ? 

39.  Six  times  seven  are  how  many  ? 

40.  Eight  times  nine  are  how  many  ? 

41.  Six  times  eight  are  how  many? 

42.  Three  times  seven  are  how  many  ? 

43.  Four  times  nine  are  how  many  ? 

44.  Three  times  eight  are  how  many? 

45.  Six  times  six  are  how  many  ? 

46.  Six  times  nine  are  how  many  ? 

47.  Nine  times  five  are  how  many  ? 

48.  Four  times  six  are  how  many  ? 

49.  Two  times  nine  are  how  many  ? 

50.  Seven  times  nine  are  how  many  ? 

51.  Nine  times  eight  are  how  many  ? 

52.  Two  times  eight  are  how  many  ? 

53.  Three  times  ten  are  how' many  ? 

54.  Eight  times  seven  are  how  many  ? 

55.  Five  times  six  are  how  many  ? 

56.  Five  times  eight  are  how  many  ? 


ect.  2.]  ARITHMETIC.  43 

57.  Two  times  seven  are  how  many  ? 

58.  Two  times  six  are  how  many  ? 

59.  Eight  times  six  are  how  many  ? 

60.  Four  times  seven  are  how  many  ? 

61.  Eight  times  eight  are  how  many  ? 
6*2.  Ten  times  five  are  how  many  ? 
6.5.  Seven  times  ten  are  how  many  ? 

64.  Ten  times  ten  are  how  many  ? 

65.  Nine  times  six  are  how  many  ? 

66.  Five  times  nine  are  how  many  ? 

67.  Three  times  nine  are  how  many? 
6<S.  Nine  times  seven  are  how  many? 

69.  Five  times  ten  are  how  many  ? 

70.  Seven  times  eight  are  how  many  ? 

71.  Five  times  seven  are  how  many  ? 

72.  Ten  times  eight  are  how  many  ? 

73.  Ten  times  seven  are  how  many? 

74.  Nine  times  ten  are  how  many  ? 

75.  Eight  times  five  are  how  many  ? 

76.  Nine  times  nine  are  how  many? 

77.  Four  times  ten  are  how  many  ? 

78.  Ten  times  six  are  how  many  ? 

79.  Eight  times  ten  are  how  many  ? 

80.  Ten  times  nine  are  how  many  ? 

D.  1.  What  cost  three  yards  of  cloth  at  five  dollars 
a  yard  ? 

2.  What  cost  four  oranges,  at  six  cents  apiece  ? 

3.  What  cost  seven  barrels  oi  cider,  at  three  dol- 
lars a  barrel  ? 

4.  How  much  do  three  barrels  of  flour  come  to,  at 
seven  dollars  a  barrel  ? 

5.  What  cost  four  firkins  of  butter,  at  eight  dollars 
H  firkin  ? 

6.  What  do  nine  pounds  of  veal  come  to,  at  six 
cents  a  pound  ? 

7.  What  cost  six  reams  of  paper,   at  five  dollars 
oer  ream 


44  ARITHMETIC.  [Part  I 

8.  What  cost  eight  pair  of  shoes,  at  three  dollars  a 
pair? 

9.  What  is  the  value  of  nine  yards  of  cloth,  at  six 
dollars  a  yard  ? 

10.  If  a  man  travel   five  miles  in   an  hour,  how 
many  miles  will  he  travel  in  nine  hours  ? 

11.  There  is  an  orchard  consisting  of  ten  rows  of 
trees,  and  nine  trees  in  each  row  ;  how  many  trees 
are  there  in  the  orchard  ? 

12.  On    a   chess-hoard    there  are   eight   rows   of 
squares,  and  eight  squares  in  each   row  ;  how  many 
squares  are  there  on  the  board  ? 

13.  In  one  penny  there  are   four  farthings  ;  how 
many  farthings  are  there  in  six  pence  ? 

14.  How  many  farthings  are  there  in  eight  pence  ? 

15.  How  many  farthings  are  there  in  nine  pence  ? 

1 6.  How  many  farthings  are  there  in  ten  pence  ? 

1 7.  In  one  shilling  there  are  twelve  pence ;  how 
many  farthings  are  there  in  a  shilling  ? 

18.  In  one  pint   there  are  four  gills  ;  how    many 
gills  are  there  in  five  pints  ? 

19.  In  one  quart  there  are  two  pints  ;  how  many 
pints  are  there  in  six  quarts  ? 

20.  How  many  pints  are  there  in  three  quarts  ? 

21.  How  many  gills  are  there  in  six  pints  ? 

22.  How  many  gills  are  there  in  one  quart  ? 

23.  How  many  gills  are  there  in  three  quarts  ? 

24.  In  one  gallon  there  are  four  quarts  ;  how  manj 
quarts  are  there  in  three  gallons  ? 

25.  How  many  quarts  are  there  in  live  gallons  ? 

26.  How  many  quarts  are  there  in  seven  gallons  ? 

27.  How  many  pints  are  there  in  one  gallon  ? 

28.  How  many  pints  are  there  in  three  gallons  ? 

29.  How  many  gills  are  there  in  one  gallon  ? 

30.  How  many  gills  are  there  in  five  'quarts  ? 
31..  How  many  gills  are  there  in  two  gallons  !' 

32.  A   person    bought    two   oranges,  at   six    cent* 


Sect.  2J  ARITHMETIC.  45 

apiece ;  and  seven  lemons,  at  four  cents  apiece ;  and 
five  pears,  at  two  cents  apiece  ;  how  much  did  the 
whole  come  to  ? 

33.  If  one  pint  of  oil  cost  eight  cents,  what  will 
one  quart  cost  ? 

34.  If  one  gill  of  alcohol  cost  four  cents,  what  will 
one  quart  cost  ? 

35.  If  one  gill  of  beer  cost   two  cents,  what  will 
one  gallon  cost  ? 

36.  If  a  stage  runs  seven  miles  in  an  hour,  how  far 
will  it  run  in  nine  hours  ? 

37.  Two  men  start  from  the  same  place  and  travel 
different  ways  ;  one  travels  two  miles  in  an  hour  ;  the 
other  travels  three  miles  in  an  hour ;  how  far  apart 
will  they  be  at  the  end  of  one  hour  ?     How  far  at 
the  end  of  two  hours  ?   How  far  at  the  end  of  three 
hours  ?  How  far  at  the  end  of  four  hours  ? 

38.  Two  men  start  from  the  same  place  and  travel 
the  same  way ;  one  travels  at  the  rate  of  two  miles  in 
an  hour ;  the  other,  four ;  how  far  apart  will  they  be 
in  one  hour  ?  How  far  in  two  hours  ?  How  far  in  four 
hours  ? 

39.  If  three  men  can  do  a  piece  of  work  in  two 
days,  how  many  days  would  it  take  one  man  to  do  it  ? 

40.  If  four  men  can  do  a  piece  of  work  in  five  days, 
how  many  days  would  it  lake  one  man  to  do  it  ? 

41.  If  six  men  can  do  a  piece  of  work  in  seven 
days,  how  marty  men  would  it  take  to  do  it  in  one 
day  ? 

42.  If  a  quantity  of  provisions  will  serve  three  men 
five  days,  how  many  men  would  it  serve  one  day  ? 

43.  If  a  quantity  of  provisions  will  serve  five  mei 
.  seven  days,  how  many  days  would  it  serve  one  man  ? 

44.  If  fifteen  dollars'  worth  of  provisions  will  serve 
eight  men  five  daysj  how  many  days  will  it  serve  one 
man  ? 

45.  A  man  had  a  piece  of  work  to  perform  which 


46  ARITHMETIC.  [Par*  I 

seven  men  could  do  in  nine  days,  but  it  was  necessary 
that  the  whole  should  be  completed  in  one  day  ;  how 
many  men  must  he  employ  ? 

46.  If  the  interest  of  one  dollar  is  six  cents  a  year 
what  would  be  the  interest  of  ten  dollars  for  the  same 
time  ? 

47.  If  the  interest  of  one  dollar  is  six  cents  for  one 
year,  what  would  be  the  interest  of  it  for  two  years  ? 
for  three  years  ?  for  seven  years  ? 

48.  If  a  man  can  earn  seven  shillings  in  a  day,  how 
many  shillings  will  he  earn  in  six  days  ? 

49.  If  a  man  can  earn  eight  dollars   in  a  month, 
how  much  can  he  earn  in  six  months  ? 

50.  At  five  dollars  a  week,  what  will  nine  weeks' 
board  come  to  ? 

51.  A  lady  bought  three  yards  of  cambric  at  two 
dollars  a  yard,  seven  yards  of  silk  for  three  dollars  a 
yard,  five  yards  of  ribbon  for  four  dollars,  and   some 
crape  for  two  dollars  ;  she  paid  four  ten-dollar  bills  ; 
how  much  must  she  receive  back  again  ? 


SECTION  HI. 

A.  1.  How  many  apples,  at  one  cent  apiece,  C*Q 
you  buy  for  four  cents  ? 

2.  How  many  pears,  at  two  cents  apiece,  can  you 
buy  for  four  cents  ? 

3.  How  many  peaches,  at  three  cents  apiece,  can 
you  buy  for  six  cents? 

4.  How  many  apples,  at  two  cents  apiece,  can  you 
buy  for  six  cents  ?     How  many  for  eight  cents  ? 

5.  How  many  pears,  at  three  cents  apiece,  can  you 
6uy  for  nine  cents  ?     How  many  for  twelve  cents  ? 

6.  If  you  have  eight  apples  to  give  to  four  boys, 
how  many  can  you  give  to  each  ? 


Sect.  3.]  ARITHMETIC.  47 

7.  If  a  man  travel  six  miles  in  two  hours,  how  many 
mile!-*  does  he  travel  in  an  hour? 

«.  11  a  man  tni  vel  three  miles  in  an  hour,  how  many 
Lours  will  it  take  him  to  travel  nine  miles? 

9.  How  many  yards  of  cloth,  at  three  dollars  a  yard, 
can  you  buy  for  fifteen  dollars  ? 

10.  If  you  had  sixteen  cents,  how  many  cakes  could 
you  buy  at  four  cents  apiece  ? 

11.  If  you  had  ten  dollars,  how  much  cloth  could 
you  buy  at  five  dollars  a  yard  ? 

1 2.  If  you  had  twelve  apples  to  give  to  six  of  your 
companions,  how  many  would  you  give  them  apiece  ? 

13.  If  a  man  can  travel  six  miles  in  an  hour,  how 
\oriff  will  it  take  him  to  travel  eighteen  miles  ? 

14.  If  a  man  can  travel  five  miles  in  an  hour,  how 
long  will  it  take  him  to  travel  twenty  miles  ? 

15.  In  a  certain  orchard  there  are  twenty-four  trees 
standing  in  rows,  and  there  are  three  trees  in  each 
row  ;  how  many  rows  are  there  ? 

16.  In  an  orchard  there  are  twenty-one  trees,  and 
there  are  seven  trees  in  each  row ;  how  many  rows 
are  there  ? 

17.  A  man  paid  twenty-seven  dollars  for  some  sheep, 
and  he  gave  nine  dollars  apiece  for  them  ;  how  many 
sheep  did  he  buy  ? 

18.  A  man  paid  twenty-eight  dollars  for  seven  bar- 
rels of  cider  ;  how  much  did  he  give  a  barrel  ? 

19.  At  five    cents    apiece,  how  many  oranges  can 
you  buy  for  thirty  cents  ? 

20.  Twenty-five  are  how  many  times  five  ? 

21.  Thirty-two  are   how  many  times  four?     How 
many  times  eight  ? 

22.  Thirty-five  are  how  many  times  seven  ?     How 
many  times  five  ? 

23.  Thirty-six  are   how   many    times    six  ?     How 
many  times  nine  ?     How  many  times  four  ? 


18  ARITHMETIC.  [Par/ 1. 

B.  Remark.  When  any  thing,  or  any  number,  is 
divided  into  two  equal  parts,  one  of  the  parts  is  called 
(he  half  of  the  thing  or  the  number. 

1.  If  an  apple  is  worth  two  cents,  what  is  one  half 
of  it  worth  ? 

2.  What  is  one  half  of  two  cents  ? 
Ans.  One  cent. 

Q.  Why? 

A.  Because,  if  you  divide  two  cents  into  two  equal 
parts,  one  of  the  parts  is  one  cent. 

3.  If  you  can  buy  a  cake  for  two  cents,  how  much 
can  you  buy  for  one  cent  ? 

4.  One  is  what  part  of  two  ? 
Ans.  One  is  the  half  part  of  two. 

5.  Two  times  one  are  how  many  times  two  ? 

6.  If  you  can  buy  one  pear  for  two  cents,  how  many 
can  you  buy  for  three  cents  ? 

7.  Three  are  how  many  times  two  ? 
Ans.  Once  two  and  one  half  of  two. 

8.  Four  are  how  many  times  two  ? 

9.  If  two  shillings  will  buy  one  yard  of  cotton  cloth, 
how  many  yards  will  five  shillings  buy  ? 

10.  Five  are  how  many  times  two  ? 
Ans.  Two  times  two  and  one  half  of  two. 

11.  'Six  are  how  many  times  two  ? 

12.  If  two  dollars  will  buy  a   yard  of  cloth,  how 
many  yards  will  seven  dollars  buy  ? 

13.  How  many  halves  make  a  whole  one? 

14.  Eight  are  how  many  times  two  ? 

15.  Nine  are  how  many  times  two? 

1 6.  Ten  are  how  many  times  two  ? 

Remark.  When  any  thing,  or  any  number,  is  divided 
into  three  equal  parts,  one  of  those  parts  is  called  the 
third  part  of  the  thing  or  number.  When  it  is  divided 
into  four  equal  parts,  one  part  is  called  the  fourth  part, 
and  so  on. 

17.  If  a  yard  of  c^oth  be  worth  three  dollars,  and 


Sect.  3.]  ARITHMETIC.  49 

it  be  cut  into  three  equal  pieces,  what  will  one  of  the 
pieces  be  worth  ?  that  is,  what  will  one  third  of  a  yard 
be  worth  ? 

18.  What  is  a  third  of  three? 

19.  Suppose  the  yard  of  cloth  to  be  cut  as  before, 
what  will  two  pieces  of  it  cost  ?  that  is,  what  will  two 
thirds  of  a  yard  cost  ? 

20.  What  is  two  thirds  of  three  ? 

21.  If  three  shillings  will  buy  one  bushel  of  corn, 
what  part  of  a  bushel  will  one  shilling  buy  ?     What 
part  of  a  bushel  will  two  shillings  buy  ? 

22.  One  is  what  part  of  three  ? 

Ans.  One  is  the  third  part  of  three  ;  or  one  third 
of  three. 

23.  Two  is  what  part  of  three  ?  . 

Ans.  Two  is  two  times  the  third  part  of  three ;  or 
two  thirds  of  three.* 

24.  Three  times  one  are  how  many  times  three  ? 

25.  If  you  can  buy  a  barrel  of  cider  for  three  dol- 
lars, how  much  can  you  buy  for  four  dollars  ?     How 
much  for  five  dollars  ? 

26.  How  many  thirds  make  a  whole  one  ? 

27.  Five  are  how  many  times  three  ? 
Ans.  Once  three  and  two  thirds  of  three. 

28.  Six  are  how  many  times  three  ? 

29.  If  you  can  buy  a  barrel  of  fish  for  three  dollars, 
how  much  can  you  buy  for  seven  dollars  ?  How  much 
for  eight  dollars  ? 

30.  What  do    you  understand  by  a  third,  and  by 
two  thirds,  of  anything  ? 

For  the  answer,  see  remark  after  example  16th. 

31.  Eighi  are  how  many  times  three  ? 

32.  Nine  are  how  many  times  three  ? 

33.  Ten  are  how  many  times  three  ? 

*  The  instructor  may  use  either  of  th>se  expressions  ;  the  for- 
mer will  be  most  easily  understood  by  the  \  upil ;  it  would  be  well, 
therefore,  for  the  instructor  to  use  the  former  frequently,  though  the 
latter  is  used  for  the  most  part  in  this  treatise. 
4 


50  ARITHMETIC.  [Part  1 

34.  Eleven  are  how  many  times  three? 

35.  Twelve  are  how  many  times  three? 

36.  If  a  yard  of  cloth  be  worth  four  dollars,  and  it 
be  'Hit  into  four  equal  parts,  what  will  one  of  the  parts 
be  worth  f    that   is,  what  is   one   fourth   of  it  worth? 
What  are  two  fourths  of  it  worth  ?    What  are  three 
fourths  of  it  worth  ? 

37.  If  you  can  buy  a  barrel  of  cider  for  four  dollars, 
how  much  can  you  buy  for  one  dollar  ?     How  much 
for  two  dollars  ?    How  much  for  three  dollars? 

38.  What  part  of  four  is  one  ? 
Ans.  One  is  the  fourth  part  of  four. 

39.  What  part  of  four  is  two  ? 
Ans.  Two  fourths  of  four. 

40.  What  part  of  four  is  three  ? 
Ans.  Three  fourths  of  four. 

41.  How  many  fourths  make  a  whole  one  ? 

42.  If  you  can  buy  a  bushel  of  corn  for  four  shil- 
lings, how  much  can  you  buy  for  five  shillings  ?     Ho\* 
much  for  six  shillings  ?     JIow  much  for  seven  shil- 
lings? 

43.  Five  are  how  many  times  four  ? 
Ans.  Once  four  and  one  fourth  of  four. 

44.  Six  are  how  many  times  four  ? 
Ans.  Once  four  and  two  fourths  of  four. 

45.  Seven  are  how  many  times  four  ? 
Ans.  Once  four  and  three  fourths  of  four. 

46.  Eight  are  how  many  times  four  ? 

47.  If  four  bushels  of  corn  will  buy  one  yard  oi 
cloth,  how  many  yards  will  nine  bushels  buy  ?     How 
many  yards  will  ten  bushels  buy  ?     How  many  yards 
will  eleven  bushels  buy  ? 

48.  What  do   you  understand  by  one  fourth,  tw 
fourths,  or  three  fourths  of  anything  ? 

See  remark  after  example  16th. 

49.  Ten  are  how  many  times  four  ? 

50.  Eleven  are  how  many  times  four  ? 


Sect.  3.]  ARITHMETIC.  51 

51.  Twelve  are  how  many  times  four  ? 

52.  Thirteen  are  how  many  times  four  ? 

53.  Fourteen  are  how  many  times  four  ? 

54.  Fifteen  are  how  many  times  four  ? 

55.  Sixteen  are  how  many  times  four  ? 

56.  If  a  barrel  of  flour  be  worth  five  dollars,  and 
it  be  divided  equally  among  five  men,  what  will  one 
man's  share  be  worth  ?  that  is,  what  is  one  fifth  of  a 
barrel  worth  ?  What  are  two  fifths  of  it  worth  ?  What 
are  three  fifths  of  it  worth  ?    What  are  four  fifths  of 
it  worth  ? 

57.  If  five  dollars  will  buy  one  box  of  butter,  what 
part  of  a  box  will  one  dollar  buy  ?     What  part  will 
two  dollars  buy  ?     What  part  will  three  dollars  buy  ? 
What  part  will  four  dollars  buy  ? 

58.  What  part  of  five  is  one  ? 
Ans.  One  is  the  fifth  part  of  five. 

59.  Two  is  what  part  of  five  ? 
Ans.  Two  fifths  of  five. 

60.  Thr  ;e  is  what  part  of  five  ? 
Ans.  Three  fifths  of  five. 

61.  Four  is  what  part  of  five  ? 

62.  How  many  fifths  make  a  whole  one  ? 

63.  If  cherries  are  five  cents  a  quart,  how  many 
quarts  can  you  buy  for  six  cents  ?     How  many  for 
seven  cents  ?     How    many    for   eight   cents  ?     How 
many  for  nine  cents  ?     How  many  for  eleven  cents  ? 
How  many  for  thirteen  cents  ? 

64.  What  do  you  understand  by  one  fifth,  two  fifths 
&c.  of  anything  ? 

See  remark  after  example  16th. 

65.  Seven  are  how  many  times  five  ? 
Ans.  Once  five  and  two  fifths  of  five  ? 

66.  Eight  are  how  many  times  five  ? 

67.  Nine  are  how  many  times  five  ? 

68.  Ten  are  how  many  times  five  ? 

69.  Elevep  are  how  many  times  five  ?„ 


52  ARITHMETIC.  [Part  1 

70.  Twelve  are  how  many  times  five  ? 

71.  Thirteen  are  how  many  times  five  ? 

72.  Fourteen  are  how  many  times  five  ? 

73.  Fifteen  are  how  many  times  five  ? 

74.  If  a  barrel  of  beef  cost  six  dollars,  and  it  were 
divided  into  six  equal  parts,  what  would  one  of  the 
parts  be  worth  ?  that  is,  what  is  one  sixth  of  it  worth  ? 
What  are  two  sixths  of  it  worth  ?     What  are  three 
sixths  of  it  worth  ?  Four  sixths  ?  Five  sixths  ? 

75.  If  fish  is  worth  six  dollars  a  ban-el,  what  part 
of  a  barrel  will  one  dollar  buy  ?    What  part  of  a  bar- 
rel will  two  dollars  buy  ?    Three  dollars  ?    Four  dol- 
lars ?  Five  dollars  ? 

76.  What  part  of  six  is  one  ? 
Am.  One  is  the  sixth  part  of  six. 

77.  What  part  of  six  is  two  ? 
Ans.  Two  sixths  of  six. 

78.  Three  is  what  part  of  six  ? 
Ans.  Three  sixths  of  six. 

79.  Four  is  what  part  of  six  ? 

80.  How  many  sixths  make  a  whole  one  ? 

81.  How  much  rye  at  six  shillings  a  bushel  can  you 
buy  for  seven  shillings  ?  How  much  for  eight  shillings  ? 
Nine  shillings  ?    Ten  shillings  ?     Eleven  shillings  ? 
Twelve  shillings  ?    Thirteen  shillings  ?     Fifteen  shil-  • 
lings  ?  Seventeen  shillings  ? 

82.  What  do  you   understand   by  one  sixth,  two 
sixths,  &c.  • 

83.  Eight  are  how  many  times  six  ? 
Ans.  One  time  six  and  two  sixths  of  six. 

84.  Nine  are  how  many  times  six  ? 

85.  Ten  are  how  many  times  six  ? 

86.  Eleven  are  how  many  times  six  ? 

87.  Twelve  are  how  many  times  six  ? 

88.  Thirteen  are  how  many  times  six  ? 

89.  Fourteen  are  how  many  times  six  ? 

90.  Fifteen  are  how  many  times  six  ? 


Sect.  3.]  ARITHMETIC.  *ft 

91.  If  coal  is  worth  seven  dollars  a  chaldron,  what 
is  one  seventh  of  a  chaldron  worth  ?  What  are  two 
sevenths    of    a    chaldron   worth  ?      Three  sevenths  ? 
Four  sevenths  ?  Five  sevenths  ?  Six  sevenths  ? 

92.  At  the  rate  of  seven  dollars  a  yard,  how  much 
broadcloth  can  you  buy  for  one  dollar  ?    How  much 
for  two  dollars  ?     How  much  for  three  dollars  ?  How 
much  for  four  dollars  ?    How  much  for  five  dollars  ? 
How  much  for  six  dollars  ?    How  much  for  eight  dol- 
lars ?    How  much  for  ten  dollars  ?     How  much  for 
twelve  dollars  ?  How  much  for  fifteen  dollars  ? 

93.  What  part  of  seven  is  one  ? 
Ans.  One  is  one  seventh  of  seven. 

94.  What  part  of  seven  is  two  ? 
Ans.  Two  sevenths  of  seven. 

95.  What  part  of  seven  is  three  ? 
9  6.  Four  is  what  part  of  seven  ? 

97.  Five  is  what  part  of  seven  ? 

98.  What  do  you  understand  by  one  seventh,  two 
sevenths,  &c.  of  anything  ? 

99.  How  many  sevenths  make- a  whole  one  ? 

100.  Nine  are  how  many  times  seven  ? 

101.  Ten  are  how  many  times  seven  ? 

102.  Eleven  are  how  many  times  seven  ? 

103.  Twelve  are  how  many  times  seven  ? 

104.  Thirteen  are  how  many  times  seven? 

105.  Fourteen  are  how  many  times  seven? 

106.  Fifteen  are  how  many  times  seven  ? 

107.  Sixteen  are  how  many  times  seven? 

108.  When  wheat  is  eight  shillings  a  bushel,  what 
is   one   eighth  of  a  bushel   worth?     What   are   two 
eighths  of  a  bushel  worth  ?      What  are  three  eighths 
of  a  bushel  worth  ?  What  are  four  eighths  of  a  bushel 
worth  ?  Five  eighths  ?  Six  eighths  ?  Seven  eighths  ? 

109.  When  wood*  is  eight  dollars  a  cord,  what  part 
jf  a   cord  can  you  buy   for  a   dollar  ?     What  part 
of  a  cord  can  you  buy  for  two  dollars  ?  What  part  for 


54  ARITHMETIC.  [Part  I. 

three  dollars  ?  What  part  for  four  dollars  ?  What  part 
for  five  dollars  ?  What  part  for  six  dollars  ?  What 
part  for  seven  dollars  ?  How  much  can  you  buy  for 
nine  dollars  ?  How  much  for  ten  dollars  ?  How  much 
for  eleven  dollars  ?  How  much  for  thirteen  dollars  ? 
How  much  for  fifteen  dollars  ?  How  much  for  nine- 
teen dollars  ? 

110.  What  part  of  eight  is  one  ? 

111.  What  part  of  eight  is  two? 

112.  Three  is  what  part  of  eight? 

113.  Four  is  what  part  of  eight? 

114.  Five  is  what  part  of  eight? 

115.  What  do  you  understand  by  one  eighth,  two 
eighths,  &c.  of  any  number  ? 

116.  Seven  js  what  part  of  eight  ? 

117.  How  many  eighths  make  a  whole  one  ? 

118.  Ten  are  how  many  times  eight  ? 

119.  Eleven  are  how  many  times  eight? 

120.  Twelve  are  how  many  times  eight? 

121.  Thirteen  are  how  many  times  eight? 

122.  Fourteen  are  how  many  times  eight? 

123.  When  sugar  is  nine  dollars  a  hundred- weight, 
what  is  one  ninth  of  a  hundred-weight  worth  ?    What 
are  two  ninths  of  a  hundred-weight  worth  ?     Three 
ninths  ?     Four  ninths  ?     Five  ninths  ?     Six  ninths  ? 
Seven  ninths  ?  Eight  ninths  ? 

124.  When  rye  is  nine  shillings  a  bushel,  what  part 
of  a  bushel  can  you  buy  for  one  shilling  ?  What  part 
for  two  shillings  ?  What  part  for  three  shillings  ?  For 
four  shillings  ?  For  five  shillings  ?  For  six  shillings  ? 
Seven  shillings  ?  Eight  shillings  ?     How  much  for  ten 
shillings  ?  For  thirteen  shillings  ?   For  fourteen  shil- 
lings ?  Sixteen  shillings  ?  Twenty  shillings  ? 

125.  What  do  you  understand  by  one  ninth,  twc 
ninths,  three  ninths,  &c.  of  any  number  ? 

126.  Three  is  what  part  of  nine  ? 

127.  Four  is  what  part  of  nine  ? 


Jftwi.  3.]  ARITHMETIC.  55 

128.  Five  is  what  part  of  nine  ? 

129.  Seven  is  what  part  of  nine? 

130.  How  many  ninths  make  a  whole  one  ? 

131.  Tim-teen  are  how  many  times  nine? 

132.  Fifteen  are  how  many  times  nine  ? 

133.  Seventeen  are  how  many  times  nine  ? 

134.  When  hay  is  ten  dollars  a  ton,  what  is  one 
tenth  of  a  ton  worth  ?  What  are  two  tenths  of  a  ton 
worth  ?  What  are  three  tenths  of  a  ton  worth  ?  Four 
tenths  ?     Five  tenths  ?     Six  tenths  ?     Seven  tenths  ? 
Eight  tenths  ?  Nine  tenths  ? 

135.  When  sugar  is  ten  dollars  a  hundred-weight, 
what  part  of  a  hundred-weight  can  you  buy  for  one 
dollar  ?    What  part  for  two  dollars  ?   What  part  for 
three  dollars  ?  What  part  for  four  dollars  ?  What  part 
for  five  dollars  ?    Six  dollars  ?  Seven  dollars  ?    Eight 
dollars  ?  Nine  dollars  ?  How  much  can  you  buy  for 
eleven    dollars  ?  Thirteen    dollars  ?   Fifteen   dollars  ? 
Seventeen  dollars  ? 

136.  What  do  you   understand  by  one  tenth,  two 
tenths,  three  tenths,  &c.  of  anything  ?  * 

137.  How  many  tenths  make  a  whole  one  ?  * 

C.  Instead  of  writing  the  names  of  numbers,  it  is 
usual  to  express  them  by  particular  characters,  called 
figures.  f 

One  is  written.     --------I' 

Two  is  written 2 

Three  is  written  --------3 

Four  is  written    --- 4 

Five  is  written  --  -  -  -  .  -  -  -  5 
Six  is  written  ..----.-6 
Seven  is  written  •  .....  7 

Eight  is  written 8 

Nine  is  written    -------     -     9 

Ten  is  written      ---.-.--10 
*  These  questions  should  frequently  be  put  to  the  pupils. 


56  ARITHMETIC.  [Part  1 

1.  Eleven  times  one  are  how  many  times  2  ? 

2.  Twelve  are  how  many  times  2  ?  3  ?  4  ? 

3.  Fourteen  are  how  many  times  2  ?  4  ?  3  ? 

4.  If  you  had  fifteen  cents,  how  many  cakes  could 
you  buy  at   4   cents  apiece  ?  How  many  at  2  cents 
apiece  ?  How  many  at  3  cents  apiece  ?  How  many  at 
5  cents  apiece  ? 

5.  Fifteen  are  how  many  times  4?  2?  3?  5? 

6.  Sixteen   are   how   many  tunes  5?   3?    6?   2? 
7?  4? 

7.  Seventeen  are  how  many  times  6?  2?  7?  3? 
5?  4? 

8.  Eighteen  are  how  many  times    4?  7?  9?   6? 
3?  2?  5?  8? 

9.  Nineteen    are   how  many  times  3?  7?  4?  5? 

8  ?  6  ?  9  ?  2  ?  10  ? 

10.  Twenty    are   how  many  times  6?  2?  8?  3? 
9?  4?  10?  5?  7? 

11.  Twenty-one   are   how  many  times  7  ?  3  ?  8  ? 
2?  4?  6?  9?  5?  10? 

12.  Twenty-two   are   how  many  times  3  ?  8  ?   5  ? 
4?9?6?7?10?2? 

13.  If  you    had   twenty-seven    dollars,  how   much 
cloth  could  you  buy  at  9  dollars  a  yard  ?  How  much 
at  6  dollars  a  yard  ?  How  much  at  4  dollars  a  yard  ? 
How  much    at    3   dollars    a    yard  ?  How  much  at  7 
dollars  a  yard  ?  How  much  at  8  dollars  a  yard  ?  How 
much  at  5  dollars  a  yard  ?  How  much  at  10  dollars 
a  yard  ? 

14.  Twenty-seven  are  how  many  times  9  ?  6  ?  4  ? 
3?  7?  8?  5?  10? 

15.  Twenty-four  are    how  many  times  6  ?  8  ?  7  ? 
5  ?  2  ?  10  ?  3  ?  4  ?  9  ? 

16.  Twenty-nine  are   how  many  tunes  3  ?  7  ?  5  ? 

9  ?  6  ?  8  ?  4  ?  10  ? 

17.  Twenty-three  are  how  many  times  4  ?  2  ?  7  ? 
8?  3?  9?  6?  5?  10? 


Sect.  3.]  ARITHMETIC.  57 

18.  Twenty-five   are    how  many  times  3  ?  7  ?  2  ? 
6?9?4?8?5?10? 

19.  Thirty  are   how  many  times    10  ?  2  ?  3  ?  7  ? 
9?  6?  5?  4?  8? 

20.  Thirty-three  are   how  many  times  6  ?  8  ?  7  ? 
4  ?  9  ?  5  ?  10  ?  3  ? 

21.  Twenty-six  are  how  many  times  9?  4?  7?  3? 
8?  5?  6?  10? 

22.  Thirty-five   are   how  many  times  5  ?   6  ?  3  ? 
7?  9?  10?  4?  8? 

23.  Thirty-eight   are   how  many  times  8  ?  6  ?  3  ? 
9?  5?  4?  7?  10? 

24.  Thirty-four   are   how  many  times    7  ?  3  ?  9  ? 
10?  6?  8?  4?  5? 

25.  Thirty-six  are  how  many  times  8?  9?  4?  5? 
3?  6?  7?  10? 

26.  Forty  are  how  many  times  8?10?6?4?3? 
9?  5?  7? 

27.  For   forty-seven  cents,  how  many  pounds   of 
meat  can  be  bought  at  6  cents  a  pound  ?  How  manj 
pounds   at  8   cents  ?  How  many  at    9    cents  ?   How 
many  at  3  cents  ?  How  many  at  5  cents.?  How  many 
at  4  cents  ?  How  many  at  7  cents  ?  How  many  at  10 
cents  a  pound  ? 

28.  Forty-seven   are    how  many  times  6  ?  8  ?  9  ? 
3?  5?  4?  7?  10? 

29.  Forty-three   are   how  many  times   9  ?  8  ?  7  ? 
6?  4?  3?  5?  10? 

30.  Forty-five  are  how  many  times  10  ?  8  ?  3  ?  6  ? 
4?  7?  5?  9? 

31.  Forty-nine    are   how  many  times  6  ?  10  ?  5  ? 
9?  4?  8?  7? 

32.  Fifty- three  are  how  many  times  8?5?6?4? 
7?  9?  10? 

33.  Fifty-seven  are  how  many  times  9  ?  7  ?  10  ? 
6  ?  5  ?  8  ?  4  ? 

*34.  Fifty-five  are  how  many  times  6?  4?  8?  10? 
9  ?  7  ?  5  ? 


58  ARITHMETIC.  [Par*  I 

35.  Forty-eight   are  how  many  times   7  ?  5  ?  9  ? 
4?  6?  8?  10? 

36.  Fifty-four  are  how  many  times  5?9?6?4? 
7?  10?  8? 

37.  Forty-four   are    how  many   times    4  ?    6  ?   9  ? 
7?  5?  8?  10? 

38.  Fifty-eight  are  how  many  times  7  ?  6  ?  8  ?  4  ? 
9?  5?  10? 

39.  Forty-six  are  how  many  times  8  ?  10  ?  1  ?  G  ? 
9?  7?  5? 

40.  Fifty  are  how  many  times  9  ?  5  ?  4  ?  10  ?  8  ? 
6?  7? 

41.  Fifty-nine  are  how  many  times  4?  8?  7?  6? 
10  ?  9  ?  5  ? 

42.  Sixty-four  are  how  many  times  7  ?  5  ?  8  ?  10  ? 
6?  9? 

43.  Sixty-eight  are  how  many  times  6  ?  8  ?  9  ?  7  ? 
10?  5? 

44.  Fifty-two  are  how  many  times  4  ?  6  ?  8  ?  10  ? 
5?  7?  9? 

45.  Sixty-three   are   how  many  times   5?  4?  6? 
10?  9?  7?  8? 

46.  Sixty-two  are  how  many  times  4  ?  10  ?  9  ?  7  ? 
8?  5?  6? 

47.  Seventy-three   are    how  many  times    10  ?    7  ? 
8?  6?  5?  9? 

48.  Seventy-five  are  how  many  times  7  ?  8  ?  10  ? 
5?  6?  9? 

49.  If   you   had    sixty-seven    dollars,    how   many 
barrels  of  flour  could  you  buy  at  5  dollars  a  barrel  ? 
How  many  at  7  dollars  a  barrel*?  How  many  at  6 
dollars  a  barrel  ?  How  many  at  8  dollars  a  barrel  ? 
How  many  at  1 0  dollars  a  barrel  ?  How  many  at  9 
dollars  a  barrel  ? 

50.  Sixty-seven  are   how  many  times   5  ?  7  ?   6  ? 
8?  10?  9? 

51.  Seventy-four  are  how  many  times  10  ?  7  ?  8  ? 
o  ?  5  ?  9  ? 


Sect.  S.]  ARITHMETIC.  59 

52.  Sixty  are  how  many  times  9  ?  10  ?  6  ?  4  ?  7  ? 
5?  8? 

53.  Seventy-two  are  how  many  times   5  ?  7  ?  6  f 
8?  10?  9? 

54.  Sixty-five  are  how  many  times  5?  10?  8?  6? 
7?  9? 

55.  Sixty-one  are  how  many  times  4?  5?  7?  6? 
8?  10?  9? 

56.  Seventy-nine  are  how  many  times  10  ?  9  ?  8  ? 
6?  5? 

57.  Seventy   are  how  many  times    9?  5?  6?  8? 
7?  10? 

58.  Eighty-two  are   how  many  times  10  ?  7  ?  8  ? 
6?  9? 

59.  Sixty-six  are  how  many  times  9?  5?  6?  7? 
10?  8? 

60.  Eighty  are   how  many  times    10?  7?  6?  8? 
9? 

61.  Sixty-nine  are  how  many  times  9?  5?  7?  10? 
8?  6? 

62.  Eighty-one  are  how  many  times    10  ?  6  ?  8  ? 
7?  9? 

63.  Seventy-six  are  how  many  tunes  9  ?  5  ?  10  ? 
6?  7?  8? 

64.  Eighty-three  are  how  many  times  10  ?  6  ?  7  ? 
9?  8? 

65.  Seventy-one   are  how  many  times   9  ?  5  ?  7  ? 
6?  8?  10? 

66.  Eighty-four  are  how  many  times  10  ?  6  ?  8  ? 
?  7  ? 

67.  Seventy-seven,  are  how  many  times  9  ?  7  ?  5 
10?  8?  6? 

68.  Eighty-five    are  how  many  times    10  ?  8  ?  7  ? 
6?  9? 

69.  Ninety  are   how  many  times   9  ?  10  ?  6  ?  7  ? 
8? 

70.  Eighty -six    are  how  many  times    10  ?  9  ?  6  ? 
7?  8? 


60  ARITHMETIC.  [Part  \ 

71.  Ninety-four  are  how  many  times  9  ?  10  ?  8  ? 
6?  7? 

72.  Eighty-seven  are  how  many  times  10  ?  9  ?  7  ? 
6?  8? 

73.  Ninety-two  are  how  many  times  9  ?  10  ?  6  ? 
7?  8? 

74.  Eighty-eight  are  how  many  times  10  ?  9  ?  8 
6?  7? 

75.  Ninety-five  are  how  many  times   9  ?  10  ?  6  ? 
8?  7? 

76.  Eighty-nine  are  how  many  times  10?  9?  6? 
7?  8? 

77.  Ninety-eight  are  how  many  times  10  ?  9  ?  8  ? 
6?  7? 

78.,  One  hundred  are  how  many  times  6  ?  10  ?  8  ? 
9?  7? 

79.  Ninety- three  are  how  many  times  10  ?  6  ?  9  ? 
8?  7? 

80.  Ninety-nine  are  how  many  times  7  ?  10  ?  8  ? 
6?  9? 

81.  Ninety-six  are  how  many  times  9?  8?  7?  6? 
10? 

82.  Ninety-seven  are  how  many  times  10  ?  9  ?  6  ? 
7?  8? 

D.  1.  If  an  orange  is  worth  3  apples,  how  many 
oranges  are  fifteen  apples  worth  ? 

2.  In  8  pints  how  many  quarts  ? 

3.  In  8  gills  how  many  pints  ? 

4.  If  you  divide  twelve  apples  equally  among  three 
»oys,  how  many  would  you  give  them  apiece  ? 

5.  How-  many  hours  would  it  take  you  to  travel 
10  miles,  if  you  travel  three  miles  in  an  hour  ? 

6.  How  many  pence  are  there  in  .eight  farthings  ? 

7.  How  many  pence  are  there  in  twelve  farthings  ? 

8.  How  many  pence  are  there  in  seventeen  far- 
things ? 


Sect.  3.]  ARITHMETIC.  61 

9.  How  many  gallons  are  there  in  ten  quarts  ? 

10.  How  much  broadcloth,  at  6  dollars  a  yard,  can 
you  buy  for  seventeen  dollars  ? 

11.  How  many  pounds  of  raisins,  at  8  cents  a  pound, 
can  you  buy  for  twenty-five  cents  ? 

12.  In  twenty-eight  shillings,  how  many  dollars  ? 

13.  In  twenty-eight  farthings,  how  many  pence  ? 

14.  How  many  barrels  of  flour,  at  7  dollars  a  bar- 
tel,  can  you  buy  for  thirty -four  dollars  ? 

15.  How  many  reams  of  paper,  at  five  dollars  a 
ream,  can  you  buy  for  thirty-seven  dollars  ? 

1 6.  In  thirty-four  gills,  how  many  pints  ? 

17.  In  twenty-seven  quarts,  how  many  gallons  ? 

18.  If  an  orange  is  worth  six  apples,  how  many 
oranges  can  you  buy  for  forty  apples  ? 

19.  Thirty-six  shillings  are  how  many  dollars  ? 

20.  A  man  bought  thirty  apples  at  the  rate  of  3  for 
a  cent,  how  many  cents  did  they  come  to  ? 

21.  A   laborer   engaged    to   work    8    months   for 
ninety-six  dollars ;  how  much  did   he  receive  for  a 
month  ?  how  much  a  week,  allowing  4  weeks  to  the 
month  ?  how  many  shillings  a  day,  allowing  6  work- 
ing-days to  the  week  ? 

22.  If  wine  is  worth  twenty  cents  a  pint,  what  is 
1  gill  worth  ? 

23.  If  you  can  buy  a  bushel  *  of  apples  for  forty 
cents,  what  is  the  price  of  a-  peck  ? 

24.  If  you  buy  a   bushel  of  pears  for  forty-eight 
cents,  what  will  be  the  price  of  half  a  peck,  at  the 
same  rate  ? 

25.  Four  men  bought  a  horse  for  forty-eight  dol- 
lars ?  what  did  each  man  pay  ? 

26.  Five    men     bought    a    horse    for   seventy-five 
dollars,  and  sold   him  again  for  forty  dollars  ;  what 
iid  each  man  lose  by  the  bargain  ? 

*  1  bushel  is  4  pecks- 


62  ARITHMETIC.  [Part  1 

27.  A  man  gave  sixty-three  cents  for  a  horse  to 
ride  nine  miles  ;  what  was  the  price  of  one  mile's 
ride? 

28.  A  man  hired  a  horse  to  ride,  and  agreed  to 
give  8    cents  a  mile  ;  he  paid  fifty-six    cents  ;  how 
many  miles  did  he  ride  ? 

29.  A  man  had    forty-two  dollars,  which   he   paid 
for  wood  at  7  dollars  a  cord  ;  how  many  cords   did 
he  buy  ? 

30.  Two  boys  are  forty-eight  rods  apart,  -and  both 
running  the  same  way  ;  but  the  hindermost  boy  gains 
upon    the    other  3  rods  in  a  minute  ;  in  how  many 
minutes  will  he  overtake  the  foremost  boy  ? 

31.  There  is  a  vessel  containing  sixty-three  gal- 
lons of  wine  ;  it  has  a  pipe  which  discharges  7  gallons 
in  an  hour  ;  how  many  hours  will  it  take  to  empty  the 
vessel ? 

32.  There  is  a  vessel  containing  eighty-seven  gal- 
lons, and  by  a  pipe  ten  gallons   will  run  into  it  in 
an    hour ;  in   how  many   hours   will   the   vessel   be 
filled  ? 

33.  If  one  man  can  do  a  piece  of  work  in  thirty 
days,  in  how  many  days  can  3  men  do  it  ?  in  how 
many  days  can  5  men  do  it  ? 

34.  If  you  wish  to  put  sixty-four  pounds  of  butter 
into  8  boxes,  how  many  pounds  would  you  put  into 
each  box  ? 

35.  If  you  had  seventy-two  pounds  of  butter,  which 
you  wished  to  put  into  boxes  containing  8  pounds  each, 
how  many  boxes  would  it  take. 

36.  If  a  man  can  perform  a  journey  in  thirty-sjx 
hours,  how  many  days  will  it  take  him  to  do  it  when 
the  days  are  nine  hours  long  ? 

37.  If  a  man  can  do  a  piece  of  work  in  forty-eight 
hours,  how  many  days  would  it  take  him  to  do  it,  if 
he  works  twelve  hours  in  a  day  ? 


Sect.  4.J  ARITHMETIC.  63 


SECTION   IV. 

A.  1.  AT  two  cents  a  yard,  what  will  3  yards  and 
one  half  of  a  yard  of  tape  cost  ? 

2.  3  times  2,  and  one  half  of  2  are  how  many  ? 

3.  At  three  dollars  a  yard,  what  will  4  yards  and 
one  third  of  a  yard  of  cloth  cost  ? 

4.  4  times  3,  and  one  third  of  3  are  how  many  ? 

5.  At  3  dollars  a  barrel,  what  will  3  barrels  and  2 
thirds  of  a  barrel  of  cider  cost  ? 

6.  3  times  3,  and  two  thirds  of  3  are  how  many  ? 

7.  If  a  man  earn  4  dollars  in  a  week,  how  many 
dollars  will   he  earn  in  3  weeks  and  1  fourth  of  a 
week  ? 

8.  3  times  4,  and  1  fourth  of  4  are  how  many  ? 

9.  If  a  yard  of  cloth   cost  4  dollars,  what  will  5 
yards  and  3  fourths  of  a  yard  cost  ? 

10.  5  times  4,  and  3  fourths  of  4  are  how  many  ? 

11.  If  a  man  spend   five  dollars  in  a  week,  how 
many  dollars  will  he  spend  in  3  weeks  and  1  fifth 
of  a  week  ?     How  much  in  5  weeks  and  2  fifths  of  a 
week  ? 

12.  3  times  5,  and  1  fifth  of  5  are  .how  many  ? 

13.  5  times  o,  and  2  fifths  of  5  are  how  many  ? 

14.  6  times  5,  and  3  fifths  of  5  are  how  many  ? 

15.  If  beer   is  worth   six    dollars  a   barrel,  what 
would  4  barrels  and  1  sixth  of  a  barrel  cost  ?     How 
much  would  7  barrels  and  5  sixths  of  a  barrel  cost  ? 

1 6.  4  times  6,  and  1  sixth  of  six  are  how  many  ? 

17.  7  times  6,  and  5  sixths  of  six  are  how  many  ? 

18.  At  7  dollars  a  barrel,  what  will  3  barrels  and 
1   seventh  of  a  barrel  of 'flour  cost?     What  will  5 
barrels  and  2  sevenths  of  a  barrel  cost  ? 

19.  3  times  7,  and  1  seventh  of  7  are  how  many? 

20.  5  times  7,  and  2  sevenths  of  7  are  how  many  I 

21.  8  times  5,  and  4  fifths  of  5  are  how  many  ? 


64  ARITHMETIC.  [Par/  1. 

22.  8  times  6,  and  3  sixths  of  6  are  how  many  ? 

23.  At  8  dollars  a  yard,  what  will  4  yards  and  1 
eighth  of  a  yard  of  broadcloth  cost  ? 

24.  4  times  8,  and  1  eighth  of  8  are  how  many  ? 

25.  2  times  7,  and  3  sevenths  of  7  are  how  many  ? 

26.  8  times  7,  and  4  sevenths  of  7  are  how  many  ? 

27.  9  times  7,  and  6  sevenths  of  7  are  how  many  ? 

28.  3  times  8,  and  5  eighths  of  8  are  how  many  ? 

29.  9  times  8,  and  7  eighths  of  8  are  how  many  ? 

30.  If  a  hundred-weight  of  sugar   cost  9   dollars, 
what  will  2  hundred-weight  and  1  ninth  of  a  hundred- 
weight cost  ?  What  will  5  hundred-weight  and  2  ninths 
of  a  hundred-weight  cost  ? 

31.  2  times  9,  and  1  ninth  of  9  are  how  many  ? 

32.  5  times  9,  and  2  ninths  of  9  are  how  many  ? 

33.  6  times  9,  and  4  ninths  of  9  are  how  many  ? 

34.  2  times  10,  and  3  tenths  of  10  are  how  many? 

35.  7  times  9,  and  7  ninths  of  9  are  how  many  ? 

36.  5  times  10,  and  4  tenths  of  10  are  how  many  ? 

37.  8  times  9,  and  5  ninths  of  9  are  how  many  ? 

38.  4  times  10,  and  7  tenths  of  10  are  how  many? 

39.  6  times  10,  and  9  tenths  of  10  are  how  many  ? 

B.  1.  A  man  bought  2  oranges  at  6  cents  apiece ; 
how  many  cents  did  they  come  to  ?  He  paid  for 
them  with  cherries  at  4  cents  a  pint ;  how  many  pints 
did  it  take  ? 

2.  2  times  6  are  how  many  times  4  ? 

3.  A  man  bought  3  yards  of  cloth  at  4  dollars  a 
yard  ;  how  many  dollars  did  it  come  to  ?     How  much 
flour  at  6  dollars  a  barrel  would  it  take  to  pay  for  it  ? 

4.  3  times  4  are  how  many  times  6  ? 

5*  A  man  bought  4  peaches  at  3  cents  apiece  ; 
how  many  cents  did  they  come  to  ?  He  paid  for  them 
with  pears  at  2  cents  apiece  ;  how  many  pears  did  it 
take? 

6.  4  times  3  are  how  many  times  2  ? 


ttcci.  4.]  ARITHMETIC.  66 

7.  Bought  2  hundred- weight  of  sugar,  at  9  dollar* 
a  hundred-weight,  and  paid  for  it  with  wood  at  6  dol- 
lars a  cord  ;  how  many  cords  did  it  take  ? 

8.  2  times  9  are  how  many  times  6  ? 

9.  Bought  3  ban-els  of  flour  at  8  dollars  a  barrel, 
and  paid  for  it  with  cider  at  4  dollars  a  barrel ;  how 
many  barrels  did  it  take  ? 

10.  3  times  8  are  how  many  times  4  ?  » 

11.  12  times  3  are  how  many  times  5  ? 

12.  6  times  4  are  how  many  times  8  ? 

13.  3  times  10  are  how  many  times  6  ? 

14.  4  times  9  are  how  many  times  6? 

15.  How   much    flannel   worth  4  shillings  a   yard 
must  be  given  for  3  yards  of  silk  worth  5  shillings 
a  yard  ? 

1 G.  3  times  5  are  how  many  times  4  ? 

17.  2  times  7  are  how  many  times  3  ?  5  ?  4  ? 

18.  4  times  5  are  how  many  times  3  ?  6  ?  7  ? 

19.  3  times  7  are  how  many  times  4?  5?  6?  8? 
9? 

20.  Bought   2   kegs  and    2    sevenths  of  a  keg  ol 
tobacco  at  7  dollars  a  keg,  and  paid  for  it  with  wood 
at  4  dollars  a  cord ;  how  many  cords  did  it  take  ? 
How  much  butter  at  3  dollars  a  box  would  it  take 
to  pay  for  it  ? 

21.  2  times  7,  and  2  sevenths  of  7  are  how  many 
times  4  ?  3  ?  5  ?  6  ?  8  ? 

22.  Bought  3  bushels  and  3  fifths  of  a  bushel  of 
corn  at  5  shillings  a  bushel,  and  paid  for  it  with  wheat 
at  6  shillings  a  bushel ;  how  many  bushels  of  wheat 
did  it  take  ? 

23.  3   times  5,  and  3  fifths  of  five  are  how  many 
times  6  ?  9  ?  4  ?  7  ?  3  >  8  ? 

24.  How  much  sugar  that  is  8  dollars  a  hundred- 
*  weight,  can  be  bought  for  4  cords  and  2  sevenths  of  a 

cord  of  wood,  at  7  dollars  a  cord  ? 


66  ARITHMETIC.  [Part  1. 

25.  4  times  7,  and  2  sevenths  of  7  are  how  many 
times  6?8?5?3?9?10? 

26.  5  times  5,  and  3  fifths  of  5  are  how  many  tunes 
4?8?9?7?10?3?6? 

27.  6  times  7,  and  3  sevenths  of  7  are  how  many 
times  9?  4?  5?  8?  10? 

28.  5  times  8,  and  3  eighths  of  8  are  how  many 
times  6  ?  9  ?  4  ?  7  ?  10  ? 

29.  7  times  8,  and  5  eighths  of  8  are  how  many 
times  9  ?  6  ?  10  ?  4  ?  5  ? 

30.  5  times  9,  and  4  ninths  of  9  are  how  many 
times  7  ?  8  ?  6  ?  10  ?  4  ? 

31.  7  times  9,  and  7  ninths  of  9  are  how  many 
times  6?  8?  10?  5?  4? 

32.  6  times  10,  and  3  tenths  of  10  are  how  many 
times  7  ?  5  ?  4  ?  9  ?  8  ? 

33.  8  tunes  10,  and  4  tenths  of  10  are  how  many 
times  6  ?  7  ?  9  ? 

34.  8  times  9,  and  3   ninths  of  9  are  how  many 
times  6  ?  10  ?  7  ? 

C.  1.  Bought  4  bushels  of  apples,  at  3  shillings  a 
bushel  *  how  many  dollars  did  they  come  to  ? 

2.  How  many  apples,  at  2  cents  apiece,  must  you 
give  for  2  lemons  at  4  cents  apiece  ? 

3.  How  many  pears,  at  3  cents  apiece,  must  you 
give  for  3  oranges  at  5  cents  apiece  ? 

4.  How  many  barrels  of  cider,  at  3  dollars  a  barrel, 
must  be  given    for  5   boxes  of  butter,  at  4   dollars 
a  box? 

5.  A  man  bought  4  yards  of  broadcloth,  at  7  dol- 
lars a  yard,  and  paid  for  it  with  flour,  at  5  dollars  a 
barrel  ;  how  many  barrels  did  he  give  ? 

6.  If  2  apples  cost  4  cents,  what  would  4  apples 
cost? 

7.  If  3  apples  are  worth  6  cents,  how  many  apples 


Sect.  4.]  AK1THMETIC.  67 

must   you  give   for  8   pears,  that  are  worth  3  cents 
apiece  ? 

8.  James  had  8  oranges  that  were  worth  5  cents 
apiece,  and  George  had  5  quarts  of  cherries  that  were 
worth  6  cents  a  quart,  which  he  gave  to  James  for 
a  part  of  his  oranges  :  how  many  oranges  did  he  buy, 
and  how  many  had  James  left  ? 

9.  Bought  8  yards  of  cloth,  at  9  shillings  a  yard 
how  many  dollars  did  it  come  to  ? 

10.  Bought  5  bushels  and  3  sevenths  of  a  bushel 
of  salt,  at  7  shillings  a  bushel ;  how  many  dollars  did 
it  come  to  ? 

11.  Bought  9  boxes  and  2  thirds  of  a  box  of  raisins 
for  3  dollars  a  box,  and  paid  for  it  with  cider  at  4  dol- 
lars a  barrel ;  how  many  barrels  did  it  take  ? 

12.  Bought   8  pounds  and  4  sevenths  of  a    pound 
of  opium  at  7  dollars  a  pound,  and  paid  for  it   with 
cloth    at   5    dollars    a  yard ;  how  many  yards  did    it 
take? 

13.  Bought  6  chaldrons  and  4  ninths  of  a  chaldron 
of  coal  at  9  dollars  a  chaldron,  and  paid  for  it  with 
oranges  at  5  dollars  a    box  ;  how  many  boxes  did  it 
take? 

14.  Bought  7  cases  and  5  sixths  of  a  case  of  Flor- 
ence oil  at  6  dollars  a  case,  and  paid  for  it  with  sheet- 
lead  at  7  dollars  a  hundred-weight ;  how  many  hun- 
dred-weight did  it  take  ? 

15.  How  many  dozen  of  eggs,  at  8  cents  a  dozen, 
must  you  give  for  7  pounds  of  sugar,  at  10  cents  a 
pound  ? 

16.  How  much  barley,  at  3  shillings  a  bushel,  must 
be  given  for  8   bushels  of  wheat,  at   7    shillings  a 
bushel  ? 

17.  How  much  cloth,  at  4  shillings  a  yard,  must  be 
given  for  a  firkin  of  butter  worth  8  dollars  ? 

18.  How  much  cloth,  at  5  shillings  a  yard,  can  be 
bought  for  2  reams  of  paper,  at  5  dollars  a  ream  ? 


68  ARITHMETIC.  [Part  1 

19.  How  much  wheat,  at  7  shillings  a  bushel,  can 
be  bought  for  2  barrels  of  cider,  at  4  dollars  and  a 
half  a  barrel  ? 

20.  How  long  would  it   take  a   man  to  lay  up  10 
dollars,  if  he  saves  4  shillings  a  day  ? 

21.  If  a  man  earns  8  shillings  a  day,  how  many 
dollars  would  he  earn  in  10  days  ? 

22.  A  man  bought  twenty  pears  at  the  rate  of  2  for 
8  cents  ;  how  much  did  they  come  to  ? 

23.  How  many  eggs,  at  the  rate  of  3  for  5  cents, 
can  you  buy  for  thirty  cents? 

24.  A  man  hired  a  laborer  and  agreed  to  give  him 
5  dollars  for  every  3  days'  work  ;  how  much  did  he 
give  him '  a  week,  there  being  6  working  days  in  a 
week  ?  How  much  was  it  a  month,  allowing  4  weeks 
to  the  month  ? 

25.  If  a  man  receives  5  dollars  for  3  days*  work, 
how  many  shillings  is  that  a  day  ? 

26.  5  men  bought  a  horse  lor  sixty-three  dollars, 
and  paid  two  dollars  a  week  for  keeping  him  ;  at  the 
end  of  8  weeks  they  sold  him  for  fifty-four  dollars ; 
how  much  did  each  man  lose  by  the  bargain  ? 


SECTION    V. 

A.  1.  JAMKS  had  4  apples,  and  John  had  half  as 
many  ;  how  many  had  he  ? 

2.  If  an  orange  cost  6  cents,  and  an  apple  half  as 
much  ;  how  much  does  the  apple  cost  ? 

3.  If  you  divide  8  apples  equally  between  two  boys, 
what  part  of  them  must  each  have  ? 

"Arts.  One  half  of  them. 

4.  What  is  1  half  of  8  ? 

5.  If  you  divide  8  apples  equally  among   4    boys 
what  part  of  them  must  each  have  ? 

Ans.  One  fourth  of  them. 


Sect.  5.]  ARITHMETIC.  69 

6.  What  is  1  fourth  of  8  ? 

7.  If  you  divide  6  oranges  equally  among  3  boys, 
what  part  of  them  must  1  boy  have  ? 

8.  What  is  1  third  of  6. 

9.  If  4  yards  of  cloth  cost  8  dollars,  what  part  of  8 
dollars  would  1  yard  cost  ?    What  part  of  8  dollars 
would  2  yards  cost  ?    What  ,part  of  8  dollars  would  3 
yards  cost  ? 

10.  What  is  1  fourth  of  8  ?     What  is  2  fourths  of 
8  ?    What  is  3  fourths  of  8  ? 

11.  If  6  yards  of  cloth  will  make  3   coats,  what 
part  of  6  yards  will  make  1  coat  ?    What  part  of  6 
yards  will  make  2  coats  ? 

12.  What  is  one  third  of  6  ?     What  is  two  thirds 
of  6? 

13.  If  3  barrels  of  cider  cost  9  dollars,1  what  part 
of  9  dollars  will  1  barrel  cost  ?     What  part  of  9  dol- 
lars will  2  barrels  cost  ? 

14.  What  is  1  third  of  9  ?  What  is  2  thirds  of  9  ? 

15.  If  2  yards  of  cloth  cost  10  dollars,  what  part 
of  10  dollars  will   1   yard  cost?  What  part  of  10  dol- 
lars will  3  yards  cost  ? 

16.  What  is  1  half  of  10  ?  What  is  3  halves  of  10  ? 

17.  If  2  barrels  of  flour  cost  twelve  dollars,  what 
part  of  twelve  dollars  will  1  barrel  cost  ?     What  part 
of  twelve  dollars  will  3  barrels  cost  ?     What  part  of 
twelve  dollars  will  5  barrels  cost  ? 

18.  What  is  1  half  of  twelve  ?     What  is  3  halves 
of  12  ?  What  is  5  halves  of  twelve  ? 

19.  If  4  barrels  of  cider  cost  twelve  dollars,  what 
part  of  twelve  dollars  will  1  ban-el  cost  ?     What  part 
of  twelve  dollars  will  3  barrels  cost  ?     What  part  of 
twelve    dollars  will    5   barrels  cost  ?     What  part  of 
twelve  dollars  will  7  barrels  cost  ? 

20.  What  is  1  fourth  of  twelve  ?  What  is  2  fourths 
of  twelve  !    3  fourths  ?    5  fourths  ?    7  fourths  ? 

21.  If  3  oranges  cost  twelve  cents,  what  part  of 


70  AKITHMETICv  [Port  1 

twelve  cents  will  1  orange  cost  ?  What  part  of  twelve 
cents  will  2  oranges  cost  ?  What  part  of  twelve  cents 
will  4  oranges  cost?  What  part  of  twelve  cents  will 
5  oranges  cost  ?  7  oranges?  10  oranges  ? 

22.  What  is  1  third  of  twelve  ?   2  thirds?   4  thirds? 

5  thirds?    7  thirds?    10  thirds? 

23.  If  5  bushels  of  wheat  cost  10  dollars,  what  part 
of  10  dollars  will   1    bushel  cost  ?     What   part  of  10 
dollars  will  2  bushels  cost?    3  bushels?    4  bushels? 

6  bushels  ?    7  bushels  ? 

24.  What   is   1  fifth  of  10?     2  fifths?     3  fifths? 

4  fifths  ?     6  iifths  ?     7  fifths  ? 

25.  What  is  1   half  of  fourteen  ?     1  seventh  ?     2 
sevenths  ?    3  sevenths  ?    5  sevenths  ? 

26.  What  is  1  third  of  fifteen?    2  thirds?    1  fifth? 

2  fifths  ?    3  fifths  ?    4  fifths  ? 

27.  What  is  1  half  of  sixteen?  3  fourths?  1  eighth? 
8  eighths  ?    5  eighths  ?    7  eighths  ?    1  sixteenth  ? 

28.  What  is  1  half  of  eighteen  ?  2  thirds?  1  sixth? 

5  sixths?    1   ninth?    2  ninths?    4  ninths?    5  ninths:' 

7  ninths  ?    8  ninths  ?    1  eighteenth  ?    5  eighteenths  ? 

29.  What  is  1  half  of  twenty  ?  1  fourth?  3  fourths? 
1  fifth?    3  fifths?'  2  fifths?    4  fifths?    1   twentieth? 

3  twentieths  ?    7  twentieths  ? 

30.  What  is  1  third  of  twenty-one  ?     2  thirds  ?     i 
seventh  r    5  sevenths  ?    3  sevenths  ?    6  sevenths  ?    4 
sevenths  ? 

31.  What  is   1   half  of  twenty-two?    1   eleventh? 

3  elevenths?   5   elevenths?   7  elevenths?    10   elev- 
enths ? 

3*2.  What  is  1  half  of  twenty-four  ?  1  third  ?  2 
thirds  ?  1  fourth  ?  3  fourths  ?  1  sixth  ?  5  sixths  ? 
5  eighths?  3  eighths?  7  eighths?  1  twelfth?  5 
twelfths?  7  twelfths? 

33.  What  is  1  half  of  twenty-six? 

34.  What  are  2  thirds  of  twenty-seven?    2  ninths? 

4  ninths  ?   5  ninths  ?   8  ninths  ?   7  ninths  ? 


Sect.  5.]  ARITHMETIC.  71 

35.  What  is  1  half  of  twenty-eight  ?    3  fourths  ?    2 
seventlis  ?    5  sevenths  ?    3  sevenths  ? 

36.  What  is  1  sixth  of  thirty?    3  fifths?  7  tenths  ? 
2  thirds?    1  half? 

37.  What  is  1  half  of  thirty-two  ?    3  fourths  ?    5 
eighths  ?    1  sixteenth  ? 

38.  What  is  1  half  of  thirty-four  ? 

39.  What  is  5  sixths  of  thirty-six  ?    2  thirds  ?    1 
half?    7  twelfths?    4  ninths?    3  fourths? 

40.  What  is  1  half  of  thirty-eight  ? 

41.  What  is  1  third  of  thirty-nine  ? 

42.  What  is  3  fourths  of  forty  ?     7  eighths  ?     9 
tenths?    1  half? 

43.  What  is  3  sevenths  of  forty-two  ?    5  sixths  ? 

B.  1.  A  boy  having  twelve  apples,  kept  1  fourth 
of  them  himself,  and  divided  the  other  3  fourths  of 
them  equally  among  4  of  his  companions,  how  many 
did  he  give  them  apiece  ? 

2.  3  fourths  of  twelve  are  how  many  times  4  ? 

3.  A  man  having  fourteen  bushels  of  grain,  divided 
5  sevenths  of  it  equally  among  3  men ;  how  much  did 
he  give  them  apiece  ? 

4.  5  sevenths  of  fourteen  are  how  many  times  3  ? 

5.  A  man  having  fifteen  shillings,  gave  away  4  fifths 
of  it,  how  many  dollars  did  he  give  away  ? 

6.  4  fifths  of  fifteen  are  how  many  times  6  ? 

7.  A  man  having  twenty-one  cents,  paid  away  6 
sevenths  of  them  for  oranges  at  5  cents  apiece  ;  how 
many  oranges  did  he  buy? 

8.  6  sevenths  of  twenty-one  are  how  many  times 
5? 

9.  A  man  bought  a  piece  of  cloth  for  twenty-four 
shillings,  and  sold  it  again  for  5  thirds  of  what  he 
gave  for  it ;  how  many  dollars  did  he  sell  it  for  ? 

10.  5  thirds  of  twenty-four  are  how  many  times  6? 

11.  7  filths  of  twenty  are  how  many  times  6  ? 


1*1  ARITHMETIC  [Part  1 

12  5  sevenths  of  thirty-five  are  how  many  times 
8? 

13.  7  sixths  of  thirty-six  are  how  many  times  5  ? 

14.  9  sevenths  of  forty-two  are  how  many  times  8! 

15.  7  ninths  of  forty-five  sire  how  many  times  6? 
10.  5  sixths  of  forty-eight  arc4  how  many  times  7? 

17.  6  fifths  of  fifty  are  how  many  times  V  ? 

18.  8  ninths  of  fifty-four  arc  how  many  times  5? 

19.  9  r^venths  of  fifty-six  are  how  many  times  10? 

20.  7  sixths  of  sixty  are  how  many  times  8  ? 

2}.  9  sevenths  of  sixty-three  are  how  many  times 
7? 

22.  10  eighths  of  sixty-four  are  how  many  times  9  ? 

23.  6  eighths  of  seVenty-two  are  how  many  times 
7? 

24.  4  sevenths  of  eighty-four  are  how  many  times 
9? 

25.  7  tenths  of  ninety  are  how  many  times  8? 

C.   ).  Charles  had  6  apples,  and  gave   1    third  of 
them  to  John  :  how  many  did  he  give  him  ? 

2.  Albert  had  9  cents,  and  spent  2  thirds  of  them  ; 
how  many  had  he  left  ? 

3.  James  had  10  pears,  and  gave  1  half  of  them  to 
one  of  his  companions,  and  2  fifths  of  them  to  another  ; 
how  many  did  he  give  away? 

4.  If  3  yards  of  cloth  cost  6  dollars,  what  is  that  a 
yard  ? 

5.  If  4  yards  of  cloth  cost  twelve  dollars,  what  will 
2  yards  cost  ? 

6.  If  9   apples   cost   eighteen    cents,   what  will 
apples  cost  ? 

7.  If  3  oranges  cost  18  cents,  what  will  2  cost  ? 

8.  James  had  twenty-five  cents,  and  he  gave  4  fifths 
Df  them  for  1 0  apples  ;  how  much  did  he  give  for  all 
the  apples  ?  how  much  apiece  ? 

9.  A  man  had  thirty  dollars,  and  gave  5  sixths  of 


SecL  5.]  ARITHMETIC.  73 

them  for  8  yards  of  cloth  ;  how  much  did  he  give  a 
yard  ? 

10.  A  man   had   forty  yards  of  cloth,  and  sold   3 
fifths  of  it  for  twenty-four  dollars  ;  what  was  that  a 
yard  ? 

11.  A  man  had  forty-two  barrels  of  flour,  and  sold 

2  sevenths  of  it  for  6  dollars  a  barrel  ;  how  much  did 
it  come  to  ? 

12.  A  boy  had  sixty-three  nuts,. and  divided  4  sev- 
enths of  them  equally  among  six  of  his  companions 
how  many  did  he  give  them  apiece  ? 

13.  If  4  yards  of  cloth  cost   twelve  dollars,  wha 
will  3  yards  cost  ? 

14.  If  5  oranges  cost  twenty-five  cents,  what  will 

3  cost? 

15.  If  3  oranges  cost  fifteen  cents,  what  will  7  cost? 

16.  If  3  barrels  of  cider  cost  twelve  dollars,  what 
will  10  barrels  cost  ? 

17.  If  7  pounds  of  flour  cost  thirty-five  cents,  what 
will  9  pounds  cost  ? 

18.  If  5  firkins  of  butter  cost  forty  dollars,  what 
will  3  firkins  cost  ? 

19.  If  2  men  can  do  a  piece  of  work  in  six  days, 
how  long  would  it  take  4  men  to  do  the  same  work  ? 

20.  If  6  men  can  do  a  piece  of  work   in   twelve 
days*  in   how   many  days  will   3   men   do  the  same 
work  ? 

21.  If  3  men  can  do  a  piece  of  work  in  twelve  days, 
in  how  many  days  will  4  men  do  the  same  work  ? 

22.  If  2  pipes  of  a  certain  size  will  empty* a  cistern 
in  6  hours,  in  how  long  a  time  will  3  pipes  of  the  same 
size  empty  it  ? 

"23.  Three  men,  setting  out  on  a  journey,  purchased 
5  loaves  of  bread  apiece,  but  before  they  had  eaten  any 
of  it,  two  other  men  joined  them,  and  they  agreed  to 
share  the  bread  equally  among  the  whole  ;  how  many 
loaves  did  they  have  apiece  ? 


74  ARITHMETIC.  *        [Part  1 

24.  If  4  barrels  of  flour  cost  twenty-four  dollars 
what  would  7  barrels  cost  ?     How  much  cider  at  3 
dollars  a  barrel  would  7  barrels  of  flour  buy? 

25.  A  man  bought  a  quantity  of  flour  for  fifty-four 
dollars  ;  and  another  man  gave  him  9  yards  of  cloth 
for  5  sixths  of  it ;  what  was  the  cloth  worth  a  yard  ? 

2G.  If  9  yards  of  cloth  cost  fifty-four  dollars ;  how 
many  boxes  of  butter,  at  4  dollars  a  box,  would  5  yards 
of  the  same  cloth  buy  ? 

27.  Bought  8  firkins  of  butter  for  seventy-two  dol- 
lars, and  gave  6  of  them  for  7  yards  of  cloth  ;  what 
was  a  yard  of  the  cloth  worth  ? 

28.  A  man  bought  6  barrels  of  flour  for  sixty  dol- 
lars, and  gave  4  barrels  of  it  for  cider  at  5  dollars  a 
Darrel ;  how  many  barrels  did  he  buy  ? 

Note.  The  manner  of  writing  numbers  with  figures 
has  been  explained  as  far  as  ten.  The  numbers  from 
ten  to  one  hundred  are  written  as  follows : 

Ten  is  written 10 

Eleven x 11 

Twelve 12 

Thirteen 13 

Fourteen  ----------14 

Fifteen 15 

Sixteen 16 

Seventeen 17 

Eighteen  ----------18 

Nineteen  ----------19 

Twenty 20 

Twenty-one 21 

Twenty-two  ---- 22 

Twenty-three 23 

Twenty-four  ---------24 

Twenty-five 25 

Twenty-six    -     - 26 

Twenty-seven     --------27 

Twenty-eight      -     -    - 28 


Sect.  6.]  ARITHMETIC.  75 

Twenty-nine 29 

Thirty 30 

Thirty-one,  &c. -     -  31 

Forty-     -   , 40 

Fifty 50 

Sixty 60 

Seventy 70 

Eighty 80 

Ninety 90 

One  hundred 100 


SECTION   VI. 

A.  1.  BOUGHT  1  half  of  a  yard  of  cloth  for  1  shil- 
ling, what  would  be  the  price  of  a  yard  at  the  same 
rate? 

2.  If  1  half  pint  of  cherries  cost  2  cents,  what  will 
a  pint  cost  ? 

3.  If  1  fourth  of  a  barrel  of  flour  cost  two  dollars, 
what  would  a  barrel  cost  ? 

4.  2  is  1  half  of  what  number  ?     1  fourth  of  what 
number  ? 

5.  If  1  third  of  a  yard  of  cloth  cost  2  dollars,  what 
is  that  a  yard  ? 

6.  2  is  1  third  of  what  number  ? 

7.  If  1  third  of  a  yard  of  cloth  cost  3  dollars,  what 
is  that  a  yard  ? 

8.  3  is  1  third  of  what  number  ? 

9.  If  1  fourth  'of  a  firkin  of  butter  cost  3  dollars, 
what  is  that  a  firkin  ? 

10.  3  is  1  fourth  of  what  number  ? 

11.  A  man  bought  1  third  of  a  bushel  of  wheat  foi 
4  shillings,  what  would  a  bushel  cost  at  that  rate  ? 

12.  4  is  1  third  of  what  number  ? 

13.  If  a  man  can  ride  2  miles  in  1  fifth  of  an  hour, 
how  far  can  he  ride  in  an  hour  ? 


76  '  ARITHMETIC.  [Par/  1 

14.  2  is  1  fifth  of  what  number  ? 

15.  A  man,  being  asked  the  age  of  his  eldest  son, 
answered  that  bis  youngest  son,  who  was  3  years  old, 
was  just  1  fifth  of  the  age  of  his  eldest  son  ;  how  old 
was  the  eldest  son  ''. 

16.  3  is  1  fifth  of  what  number? 

17.  A  man  bought  1  sixth  part  of  a  hundred-weigh! 
of  sugar  for  2  dollars,  what  would  a  hundred-weight 
cost  at  the  same  rate  ? 

18.  2  is  1  sixth  of  what  number? 

19.  Bought  1  fifth  of  a  pound  of  starch  for  5  cents, 
what  was  that  a  pound  ? 

20.  5  is  1  fifth  of  what  number  ? 

21.  Bought  1  fourth  of  a  pound  of  aquafortis  for  6 
cents,  what  was  that  a  pound  ? 

22.  6  is  1  fourth  of  what  number  ? 

23.  Bought  1  seventh  of  a  hundred-weight  of  cocoa 
for  4  dollars,  what  would  1  hundred- weight  cost  at  the 
same  rate  ? 

24.  4  is  1  seventh  of  what  number  ? 

25.  7  is  1  fifth  of  what  number  ? 

26.  5  is  1  third  of  what  number  ? 

27.  4  is  1  eiirhth  of  what  number  ? 

28.  6  is  1  sixth  of  what  number  ? 

29.  8  is  1  third  of  what  number  ? 

30.  9  is  1  fourth  of  what  number  ? 

31.  7  is  1  sixth  of  what  number  ? 

32.  8  is  1  seventh  of  what  number  ? 

33.  9  is  1  eighth  of  what  number  ? 

34.  8  is  1  tenth  of  what  number? 

35.  7  is  1  ninth  of  what  number  ? 

36.  6  is  1  fifth  of  what  number  ? 

37.  10  is  1  seventh  of  what  number  ? 

B.  1.  A  man  bought  some  linen  and  some  cotton 
cloth  :  for  the  linen  he  gave  4  shillings  a  yard,  which 
was  twice  as  much  as  he  gave  for  the  cotton  ;  what 
did  he  give  a  yard  for  the  cotton  ? 


feet.  6.]  ARITHMETIC.  77 

/» 

2.  4  is  2  times  what  number  ? 

3.  If  2  thirds  of  a  yard  of  cloth   cost    6   dollars 
what  would  1  third  cost  ?  _ 

4.  6  is  2  times  what  number  ? 

5.  If  3  fourths  of  a  barrel  of  flour  cost  6  dollars, 
what  will  1  fourth  of  a  barrel  cost  ? 

6.  6  is  three  times  what  number  ? 

7.  If  2  fifths  of  a  pound  of  chocolate  cost  8  cents, 
what  would  1  fifth  of  a  pound  cost  ? 

•8.  8  is  2  times  what  number  ? 

9.  If  3  fifths  of  a  pound  of  candles  cost  9  cents, 
what  will  1  fifth  of  a  pound  cost  ? 

10.  9  is  3  times  what  number  ? 

11.  If  2  sevenths  of  a  pound  uf  spermaceti  candles 
cost  10  cents,  what  will  1  seventh  of  a  pound  cost  ? 

12.  10  is  2  times  what  number? 

13.  If  5  eighths  of  a  pound  of  cotton  cost  10  cents, 
what  will  1  eighth  cost  ? 

14.  10  is  5  times  what  number  ? 

15.  If  2  thirds  of  a  yard  of  cloth   cost  4 'dollars, 
what  will  1  third  cost  ?     If  1  third  of  a  yard  cost  2 
dollars,  what  will  a  yard  cost  ?  * 

16.  If  4  is  2  thirds  of  some  number,  what  is    1 
third  of  the  same  number  ?  —  2  is  1  third  of  what  num- 
ber ?  *  —  Then  4  is  2  thirds  of  what  ? 

17.  If  2  thirds  of  a  barrel  of  flour  cost  6  dollars, 
what  will  1  third  of  a  barrel  cost  ?     If  1  third  of  a 
barrel  cost  3  dollars,  what  will  a  barrel  cost  ?  * 

18.  If*  6  is  two  thirds  of  some  number,  what  is  1 
third  of  the  same   number  ?  —  3  is   1   third  of  what 
number  ?  —  Then^  6  is  2  thirds  of  what  ? 

19.  If  3  fourths  of  a  bushel  of  wheat  cost  6  shil- 
lings, what   will    1    fourth    of  a    bushel    cost  ?     If  1 
fourth  of  a  bushel  cost  2  shillings,  what  will  a  busheJ 
cost  ? 

20.  If  6  is  3  fourths  of  some  number,  what  is  1 

*  See  this  section,  article  A. 


78  ARITHMETIC.  [Part  1. 

fourth  of  the  same  number  ?  —  2  is  1  fourth  of  what 
number  ?  —  Then  6  is  3  fburths  of  what  ? 

21.  If  2  fifths  of  a  gallon  of  wine  cost  4  shillings, 
what  will  1  fifth  of  a  gallon  cost  ?    If  1  fifth  of  a  gal- 
lon cost  2  shillings,  what  will  a  gallon  cost  ? 

22.  If  4  is  2  fifths  of  some  number,  what  is  1  fifth 
of  the  same  number  ?  —  2  is  1  fifth  of  what  number  ? 
—  Then  4  is  2  fifths  of  what  ? 

23.  If  3   sevenths   of  a   pound  of  tobacco  cost    6 
cents,  .what  will  1  seventh  of  a   pound   cost?     If  1 
seventh  of  a  pound  cost  2  cents,  what  will  a  pound 
cost? 

24.  If  6  is  3  sevenths  of  some  number,  what  is  1 
seventh  of  the  same   number  ?  —  2  is  1  seventh  of 
what  number  ?  —  Then  6  is  3  sevenths  of  what  ? 

25.  If  2  sevenths  of  a  barrel  of  fish  cost  4  dollars, 
what  will   1   seventh  of  a   barrel   cost  ?     What  will 
a  barrel  cost  ? 

26.  4  is  2  sevenths  of  what  number  ? 

27.  If  3  eighths  of  a  pound  of  chocolate  cost  6  cents, 
what  will  1  eighth  of  a  pound  cost?     What  will  a 
pound  cost  ? 

28.  6  is  3  eighths  of  what  number  ? 

29.  If  eight  cents  will  buy  2  fifths  of  a  pound  of 
aquafortis,  how  many  cents  will  buy  a  pound  ? 

30.  8  is  2  fifths  of  what  number  ? 

31.  A  man  bought -3  fourths  of  a  hundred- weight  of 
yellow  ochre  for  9  dollars  ;  what  was  that  a  hundred- 
weight ? 

32.  9  is  3  fourths  of  what  number  ? 

33.  8  is  4  ninths  of  what  number  ? 

34.  9  is  3  tenths  of  what  number  ? 

35.  10  is  5  sevenths  of  what  number  ? 

36.  12  is  3  fifths  of  what  number  ? 

37.  12^  is  4  ninths  of  what  number  ?. 

38.  10'is  2  sevenths  of  what  number? 

39.  14  is  7  fifths  of  what  number  ? 


Sect.  6.]  ARITHMETIC.  79 

40.  15  is  3  elevenths  of  what  number  ? 

41.  16  is  2  fifths  of  what  number? 

42.  18  is  6  tenths  of  what  number? 

43.  20  is  5  ninths  of  what  number  ? 

44.  21  is  3  ninths  of  what  number  ? 

45.  25  is  8  ninths  of  what  number  ? 

C.  1.  If  5  eighths  of  a  cask  of  claret  wine  cost  15 
dollars,  what  is  that  a  cask  ?  How  much  cider  at  6 
dollars  a  Barrel  would  it  take  to  pay  for  a  cask  of  the 
wine  ? 

2.  15  is  5  eighths  of  how  many  times  4  ? 
,    3.  If  2  thirds  of  a  pound  of  coffee  cost  18  cents, 
how  much  would  a  pound  cost  ?     How  many  oranges 
at  5  cents  apiece,  might  be  bought  for  a  pound  ? 

4.  18  is  2  thirds  of  how  many  times  5  ? 

5.  A  man  bought  4  sevenths  of  a  hundred-weight  of 
sugar  for  20  shillings,  how  many  dollars  would  a  hun- 
dred-weight come  to  at  the  same  rate  ? 

6.  20  is  4  sevenths  of  how  many  times  6  ? 

7.  A  man  sold  a  cow  for  21  dollars,  which  was  only 
7  tenths  of  what  she  cost  him  ;  how  much  did  she  cost 
him  ?     When  he  bought  her,  he  paid  for   her  with 
cloth  at  8  dollars  a  yard ;  how  many  yards  of  cloth 
did  he  give  ? 

8.  21  is  7  tenths  of  how  many  times  8  ? 

9.  A  man,  being  asked  the  age  of  his  youngest  son, 
answered,  that  the  age  of  his  eldest  son  was  24  years, 
which  was  3  fifths  of  his  own  age  ;  and  that  his  own 
age  was  10  times  as  much  as  that 'of  his  youngest 
Bon  ;  what  was  his  age  ?  and  what  was  the  age  of  hia 
youngest  son  ? 

,    10.  24  is  3  fifths  of  how  many  times  10  ? 

11.  27  is  3  fifths  of  how  many  times  7  ? 

12.  28  is  7  tenths  of  how  many  times  9  ? 

13.  30  is  5  eighths  of  how  many  times  7  ? 

14.  32  is  4  sevenths  of  how  many  times  6? 


80  ARITHMETIC.  [Part  1 

15.  36  is  9  eighths  of  how  many  times  5  ? 

1 6.  40  is  8  ninths  of  how  many  times  8  ? 

17.  42  is  6  fifths  of  how  many  times  4  ? 

18.  45  is  9  eighths  of  how  many  times  6  ? 

19.  48  is  8  ninths  of  how  many  times  7  ? 

20.  50  is  5  sevenths  of  how  many  times  8  ? 

21.  54  is  9  sixths  of  how  many  times  7  ? 

22.  56  is  7  ninths  of  how  many  times  10  ? 

23.  60  is  10  sevenths  of  how  many  times  4? 

24.  63  is  9  eighths  of  how  many  times  5  ? 

25.  64  is  8  ninths  of  how  many  times  7  ? 

26.  70  is  10  sevenths  of  how  many  times  8  ? 

27.  72  is  9  fifths  of  how  many  times  6  ? 

28.  80  is  10  thirds  of  how  many  times  4  ? 

29.  80  is  8  fifths  of  how  many  times  6  ? 

D.  1.  A  boy  gave  away  4  cents,  which  was  1  third 
of  all  he  had  ;  how  many  had  he  at  first  ? 

2.  A  boy  gave  5  apples  to  one  of  his  companions, 
which  was  1  fourth  of  what  he  had ;  how  many  had 
he? 

3.  A  man  paid  away  4  dollars,  which  was  2  thirds 
ot  all  the  money  he  had  ;  how  much  had  he  ? 

4.  A  man  sold  a  watch  for  18  dollars,  which  was  3 
fourths  ef  what  it  cost  him  ;  how  much  did  it  cost  ? 

5.  A  man  sold  a  cow  for  15  dollars,  which  was  3 
fifths  of  what  the  cow  cost ;  how  much  did  he  lose  by 
his  bargain  ? 

6.  A  man  bought  12  yards  of  cloth,  and  sold  it  for 
54  dollars,  which  was  9  eighths  of  what  it  cost  him ; 
what  did  it  cost  him  a  yard  ?  and  how  much  did  he 
gain  by  his  bargain  ? 

7.  There  is  a  pole  standing  in  the  water,  so  that  10 
feet  of  it  is  above  the  water,  which  is  2  thirds  of  the 
whole  length  of  the  pole  ;  how  long  is  the  pole  ? 

8.  There  is  a  pole  2  thirds  under  water,  and  4  feet 
mt ;  how  long  is  the  pole  ? 


Sect.  6.]  ARITHMETIC.  81 

9.  There  is  a  pole  2  fifths  under  water,  and  6  feet 
out  of  the  water  ;  how  long  is  the  pole  ? 

10.  There  is  an  orchard,  in  which  3  sevenths  of  the 
trees  bear  cherries,  and  2  sevenths  bear  peaches,  and 
10  trees  bear  plums  ;  how  many  trees  are  there  in  the 
orchard  ?  and  how  many  of  each  sort  ? 

11.  There    is   a  school,  in  which    2  ninths  of  the 
boys  learn  arithmetic,  3  ninths  learn  grammar,  I  ninth 
learn  geography,  1  ninth  learn  geometry,  and  1 2  learn 
to  write  ;  how  many  are  there  in  the  school  ?  and  how 
many  attending  to  each  study  ? 

12.  A  man  sold  a  watch  for  63  dollars,  which  was 
7  fifths  of  what  it  cost  him ;  how  much  did  he  gain 
by  the  bargain  ? 

Miscellaneous  Examples. 

1.  IF  1  yard  of  cloth  cost  4  dollars,  what  will  5 
yards  cost  ? 

2.  A  man  bought  3  pounds  of  raisins,  at  7  cents 
a    pound,  and  16  oranges,  at  4  cents  apiece,  and  1 
pound  of  candles   for   1 6    cents  ;  what   did   they  all 
come  to  ? 

3.  A  boy  had  37  apples ;  he  kept  5  himself,  and 
divided  the  rest  equally  among  four  companions  ;  how 
many  did  he  give  them  apiece  ? 

4.  Two  men  are  40  miles  apart,  and  both  travelling 
the  same  way  ;  the  hindermost  man  gains  upon  the 
other  5  miles  each  day ;  in  how  many  days  will  he 
overtake  him  ? 

5.  Two   men   are   travelling   the  same  way ;  one 
travels  at  the  rate  of  38  miles  a  day,  the  other,  who 
is  behind   the  former,  travels  44   miles ;  how  much 
does  he  gain  of  the  first  each  day  ?  and  in  how  many 
days  would  he  gain  60  miles  ? 

1    6.  A  fox  is  80   rods  before  a  greyhound,  and  is 
running  at  the  rate  of  27  rods  in  a  minute  ;  the  grey- 


82  ARITHMETIC.  [Par/  1, 

hound  is  following  at  the  rate  of  31  rods  in  a  minute  ^ 
in  how  many  minutes  will  the  greyhound  overtake 
the  fox. 

7.  If  2  yards  of  cloth  cost  6  dollars,  what  would  4 
yards  cost?  what  would  12  yards  cost  ? 

8.  If  8  sheep  cost  24  dollars,  what  would  3  cost  ? 
i).  if  4  tons  of  hay  will  keep  3  horses  through  the 

winter,  how  many  tons  will  keep  30  horses  the  same 
time  ? 

10.  If  a  man  spends  8  shillings  a  day,  how  many 
dollars  will  he  spend  in  a  week  ? 

11.  Bought  ten  pieces  of  cloth,  each  containing  5 
yards,  for  100  dollars  ;  what  was  it  a  piece  ?  and  how 
much  a  yard  ? 

12.  If  the  wages  of  12  weeks  come  to  60  dollars, 
what  is  that  a  month  ?  and  how  much  for  5  weeks. 

13.  If  7  horses  eat  14  bushels  of  oats  in  1  week, 
how  many  bushels  would  15  horses  eat  in  the  same 
time  ? 

14.  If  3  horses  eat  8  bushels  of  oats  in  2  weeks, 
how  long  would  it  take  them  to  eat  40  bushels  ? 

15.  If  1  horse  eat  1  bushel  of  oats  in  3  days,  in 
how  many  days  would  4  horses  eat  36  bushels  ? 

16.  If  2  men  spend   12   dollars   in   1  week,  how 
many  dollars  would  3  men,  at  the  same  rate,  spend  in 
5  weeks  ? 

17.  If  a  staff  3  feet  long  cast  a  shadow  of  2  feet 
Ht  12  o'clock,  what  is  the  length  of  a  pole  that  casts 
a  shadow  18  feet  at  the  same  time  of  day  ? 

18.  If  47  gallons  of  water,  in   I  hour,  run  into  a 
cistern  containing  108  gallons,  and  by  a  pipe  38  gal- 
lons run  out  in  an  hour,  how  much  remains  in  the 
cistern  in  an  hour  ?   and  in  how  many  hours  will  the 
cistern  be  filled  ? 

19.  If  4  men  can  do  a  piece  of  work  in  8  days, 
how  many  men  would  it  take  to  do  the  same  work  in 
4  days  ? 


Sect.  6.]  ARITHMETIC.  83 

20.  If  6  men  can  do  a  piece  of  worfc  in  9  days,  in 
how  many  days  would  2  men  do  it  ? 

21.  If  8  men  can  do  a  piece  of  work  in  5  days, 
in  how  many  days  would  they  do  a  piece  of  work  4 
times  as  large  ? 

22.  If  7  men  can  do  a  piece  of  work  in  3  days, 
how  many  men  would  it  take  to  do  a  piece  of  work  5 
times  as  large  in  the  same  time  ? 

23.  If  8  men  can  do  a  piece  of  work  in  4  days, 
in  how  many  days  would  2  men  do  a  piece  of  work 
one  half  as  large  ? 

24.  A  man  bought  a  cask  of  wine  containing  63 
gallons,  3  sevenths  of  which  leaked  out ;  and  he  sold 
the  remainder  for  36  dollars  ;  how  much  per  gallon 
did  he  sell  it  for  ? 

25.  If  a  cask  of  wine  cost  48  dollars,  what  is  5 
eighths  of  it  worth  ? 

26.  A  man  bought  7  oranges  for  6  cents  apiece, 
and  sold  them  all  for  54  cents,  how  much  did  he  gain 
by  the  bargain  ? 

27.  A  man  bought  8  yards  of  cloth  for  56  dollars, 
and  sold  it  again  for  9  dollars  a  yard  ;  how  much  did 
he  gain  by  the  bargain  ? 

28.  A  man  bought  8  barrels  of  flour  for  40  dollars  ; 
how  much  must  he  sell  it  at  per  barrel  to  gain  16 
dollars  ? 

29.  A  man  bought  5  firkins  of  butter  at  7  dollars 
a  firkin  ;  how  much  must  he  sell  it  per  firkin  to  gain 
10  dollars  ? 

30.  A  man  gave  35  cents  for  his  breakfast,  which 
was  5  eighths  of  what  he  gave  for  his  dinner ;  what 
did  he  give  for  his  dinner  ? 

31.  A  ship's  crew  of  6  men  have  provision  for  3 
months  ;  how  many  months  would  it  last  1  man  ? 

32.  A  ship's  crew  have  provision  sufficient  to  last  1 
oian  27  months  ;  how  long  would  it  last  9  men  ? 

33.  A   ship's   crew    have    provision    sufficient    to 


84  ARITHMETIC.  [Par/  1. 

last  3  men    10   months  ;  how  long  would   it   last   5 
men  ? 

34.  A  man  built  40  rods  of  wall  in  a  certain  time, 
another  man  can  build  9  rods  while  the  first  builds  5 ; 
how  much  would  he  build  in  the  same  time  ? 


SECTION   VIL 

If  the  combinations  in  this  section  should  be  found 
too  difficult,  they  may  be  omitted  until  reviewing  the 
l>ook. 

A.  1.  A  MAN  being  asked  the  age  of  his  eldest  son, 
answered,  that  his  youngest  son  was  six  years  old, 
and  that  2  thirds  of  the  youngest  son's  age  was  just  1 
fifth  of  the  eldest  son's  age.  Required  the  age  of 
the  eldest  son. 

2.  2  thirds  of  6  is  1  fifth  of  what  number  ? 

3.  A  man,  being  asked  how  many  sheep  he  had, 
said  that  he  had  them  in  two  pastures  ;  in  one  pasture 
he  had  eight ;  and  that  3  fourths  of  these  was  just  1 
third  of  what  he  had  in  the  other.     How  many  were 
there  in  the  other  ? 

4.  3  fourths  of  8  is  1  third  of  what  number  ? 

5.  Two  boys  talking  of  their  ages,  one  said  he  was 
9  years  old.     Well,  said  the  other,  2  thirds  of  your 
age  is  exactly  3  fourths  of  my  age  ;  now,  if  you  will 
tell  me  how  old  I  am,  I  will  give  you  as  many  apples 
as  I  am  years  old.     What  was  his  age  ? 

6.  2  thirds  of  9  is  3  fourths  of  what  number  ? 

7.  Two  boys  counting  their  money,  one  said  he  had 
ten  cents.     The  other  says,  4  fifths  of  your  money  is 
exactly  2  sevenths  of  mine  ;  now,  if  you  will  tell  how 
many  I  have,  I  will  give  you  1  half  of  them.     How 
many  had  he  ? 

8.  4  fifths  of  10  is  2  sevenths  of  what  number  r 


Sect  7.]  ARITHMETIC.  85 

9.  5  sixths  of  12  is  2  thirds   of  what  number  ? 

10.  6  sevenths  of  14  is  4  ninths  of  what  number? 

11.  '6  filths  of  15  is  2  thirds  of  what  number? 

12.  7  ninths  of  18  is  2  fifths  of  what  number  ? 

13.  4  fifths  of  20  is  8  sevenths  of  what  number  ? 

14.  8  sevenths  of  21  is  6  tenths  of  what  number? 

15.  5    sixths    of  24   is    10  sevenths  of  how  many 
times  5  ? 

16.  3   sevenths  of  28    is  2   eighths  of  how  many 
times  7? 

17.  4  fifths  of  30  is  6  sevenths  of  how  many  times 
8? 

18.  6  eighths  of  32  is  8  ninths  of  how  many  times 
5? 

19.  4  ninths  of  36  is  8  tenths  of  how  many  times 
6? 

20.  3   fourths   of  40  is  5    sevenths  of  how  many 
times  8  ? 

21.  6  ninths  of  45  is  3  fifths  of  how-  many  times 
7? 

22.  5  sixths  of  48   is    10  sevenths   of  how  many 
times  3? 

23.  4  sevenths  of  63  is  6  fifths  of  how  many  times 
8? 

24.  5  ninths  of  72  is  4  sevenths  of  how  many  times 
9? 

B.  1.  4  fifths  of  15  is  6  tenths  of  how  many  thirds 
of  21? 

2.  4  thirds  of  18  is  8  ninths  of  how  many  sevenths 
of  35  ? 

3.  6  sevenths  of  21  is  2  thirds  of  how  many  thirds 
of  24  ? 

4.  5   fourths  of  24    is    10  sevenths  of  how  many 
fifths  of  40. 

5.  5  eighths  of  32   is  2  fifths  of  how  many  fifths 
tf  35? 


86  ARITHMETIC  [Part  \ 

6.  4   sevenths   of  63   is   6   eighths   of  how  many 
ninths  of  45  ? 

7.  3   sevenths    of  56    is    4   ninths   of  how  many 
fourths  of  28  ? 

8.  3  eighths  of  64  is  6  tenths  of  how  many  sixths 
of  30. 

9.  2  eighths  of  72  is  3  tenths  of  how  many  til'tha 
of  40? 

C,  1.  Two  times  eleven  are  how  many  ? 

2.  Two  times  twelve  are  how  many  ? 

3.  Two  times  thirteen  are  how  many  ? 

4.  Two  times  fourteen  are  how  many  ? 

5.  Two  times  fifteen  are  how  many  ? 

6.  Two  times  sixteen  are  how  many  ? 

7.  Two  times  seventeen  are  how  many  ? 

8.  Two  times  eighteen  are  how  many  ? 

9.  Two  times  nineteen  are  how  many  ? 

10.  Two  times  twenty  are  how  many  ? 

11.  Three  times  eleven  are  how  many  ? 

12.  Three  times  twelve  are  how  many  ? 

13.  Three  times  thirteen  are  how  many  ? 

1 4.  Three  times  fourteen  are  how  many  ? 

15.  Three  times  fifteen  are  how  many  ? 

16.  Three  times  sixteen  are  how  many  ? 

17.  Three  times  seventeen  are  how  many  ? 

1 8.  Three  times  eighteen  are  how  many  ? 

19.  Three  times  nineteen  are  how  many  ? 

20.  Three  times  twenty  are  how  many  ? 

21.  Four  times  eleven  are  how  many  ? 

22.  Four  times  twelve  are  how  many  ? 

23.  Four  times  tliirteen  are  how  many  ? 

24.  Four  times  fourteen  are  how  many  ? 

25.  Four  times  fifteen  are  how  many  ? 

26.  Four  times  sixteen  are  how  many? 

27.  Four  times  seventeen  are  how  many 

28.  Four  times  eighteen  are  how  many  ? 


Sect.  7  ]  ARITHMETIC.  87 

29.  Four  times  nineteen  are  how  many  '< 

30.  Four  times  twenty  are  how  many  ? 

31.  Five  times  eleven  are  how  many  ? 

32.  Five  times  twelve  are  how  many  ? 

33.  Five  times  thirteen  are  how  many  ? 

34.  Five  times  fourteen  are  how  many  ? 

35.  Five  times  fifteen  are  how  many  ? 
36    Five  times  sixteen  are  how  many  ? 

37.  Five  times  seventeen  are  how  many  ? 

38.  Five  times  eighteen  are  how  many  ? 

39.  Five  times  nineteen  are  how  many  : 

40.  Five  times  twenty  are  how  many  ? 

41.  Six  tinges  eleven  are  how  many  ? 

42.  Six  times  twelve  are  how  many  ? 

43.  Six  times  thirteen  are  how  many  ? 

44.  Six  times  fourteen  are  how  many  ? 

45.  Six  times  fifteen  are  how  many  ? 

46.  Six  times  sixteen  are  how  many  ? 

47.  Six  times  seventeen  are  how  many  ? 

48.  Six  tunes  eighteen  are  how  many  ? 

49.  Six  times  nineteen  are  how  many  ? 

50.  Six  times  twenty  are  how  many  'i 

51.  Seven  times  eleven  are  how  many  ? 

52.  Seven  times  twelve  are  how  many  ? 

53.  Seven  times  thirteen  are  how  many  ? 

54.  Seven  times  fourteen  are  how  many  ? 

55.  Seven  times  fifteen  are  how  many  ? 

56.  Seven  times  sixteen  are  how  many  ? 

57.  Seven  times  seventeen  are  how  many  ? 

58.  Seven  times  eighteen  are  how  many  ? 

59.  Seven  times  nineteen  are  how  many  ? 

60.  Seven  times  twenty  are  how  many  ? 

61.  Eight  times  eleven  are  how  many  ? 

62.  Eight  times  twelve  are  how  many  ? 

63.  Eight  times  thirteen  are  how  many  ? 

64.  Eight  times  fourteen  are  how  many  ? 

65.  Eight  times  fifteen  are  how  many  ? 


88  ARITHMETIC.  [Part  ) 

66.  Eight  times  sixteen  are  how  many  ? 

67.  Eight  times  seventeen  are  how  many  ? 

68.  Eight  times  eighteen  are  how  many  ? 

69.  Eight  times  nineteen  are  how  many  ? 

70.  Eight  times  twenty  are  how  many  v 

71.  Nine  times  eleven  are  how  many  ? 

72.  Nine  times  twelve  are  how  many  ? 

73.  Nine  tunes  thirteen  are  how  many  ? 

74.  Nine  times  fourteen  are  how  many  ? 

75.  Nine  times  fifteen  are  how  many  ? 

76.  Nine  times  sixteen  are  how  many  ? 

77.  Nine  times  seventeen  are  how  many  ? 

78.  Nine  times  eighteen  are  how  many  ? 

79.  Nine  tunes  nineteen  are  how  in  any  ? 

80.  Nine  times  twenty  are  how  many  ? 

81.  Ten  times  eleven  are  how  many  ? 

82.  Ten  times  twelve  are  how  many  ? 

83.  Ten  times  thirteen  are  how  many  ? 

84.  Ten  times  fourteen  are  how  many  ? 

85.  Ten  times  fifteen  are  how  many  ? 

86.  Ten  times  sixteen  are  how  many  ? 

87.  Ten  tunes  seventeen  are  how  many  ? 

88.  Ten  times  eighteen  are  how  many  ? 

89.  Ten  times  nineteen  are  how  many  ? 

90.  Ten  times  twenty  are  how  many  ? 


SECTION  VIU. 

A.  1.  IF  you  cut  an  apple  into  two  equal  parts., 
what  is  one  of  those  parts  called  ?  * 

2.  How  many  halves  of  an  apple  will  make  the 
whole  apple  ? 

3.  If  you  cut  an  apple  into  3  equal  parts,  what  is 

*  See  Section  III.,  article  B,  remark  before  questions  1  axid  17 


Sect.  8  j  ARITHMETIC  89 

1   of  those   parts    called  ?    what   are    2  of   the    parts 
called  ? 

4.  How  many  thirds  of  an   apple   will    make  the 
whole  apple  ? 

5.  If  you  cut  an  apple  into  4  equal  parts,  what  is  I 
of  those   parts   called?    what   are    2    of  those   parta 
called  ?    what  are  3  of  them  called  ? 

6.  How  many  fourths  of  an  apple  make  the  whole 
apple  ? 

7.  If  an  apple  be  cut  into  5  equal  parts,  what  is 
1  of  the  parts  called  ?  what  are  2  of  the  parts  called  ? 
what  are  3  of  the  parts  called  ?    what  are  4  of  the 
parts  called  ? 

8.  How  many  fifths  of  an  apple  make  the  whole 
apple  ? 

9.  If  an  apple  be  cut  into  6  equal  parts,  what  is 
1  of  the  parts  called  ?  what  are  2  of  the  parts  called  ? 
what  3  ?  what  4  ?  what  5  ? 

10.  How  many  sixths  of  an  apple  make  the  whole 
apple  ? 

11.  If  an  apple  be  cut  into  7  equal  parts,  what  is  1 
of  the  parts  called  ?    what  are  2  of  the  parts  called  ? 
what  3  ?  what  4  ?  what  5  ?  what  6  ? 

Let  the  instructor  ask  the  pupils  the  divisions  of  a 
unit  in  this  manner  as  far  as  the  division  into  10  parts 
It  would  be  well  to  ask  them  further.  Then  let  him 
begin  again,  and  suppose  an  orange  instead  of  an  ap- 
ple. After  applying  the  division  to  several  different 
thing  Si  a  line  may  be  drawn  on  the  blackboard  and 
divided.  It  is  well  often  to  require  the  pupil  to  make 
the  requisite  division  of  a  line  on  the  board. 

12.  A  man  had  a  bushel  of  corn,  and  wished  to 
give  1  half  of  a  bushel  apiece  to  some  laborers  ;  bow 
many  could  he  give  it  to  ? 

13.  How  many  halves  are  there  in  1  ? 


90  ARITHMETIC.  [/'art  1 

14.  A  man  divided  2  barrels  of  flour  among  his  la- 
borers, giving  them  1   half  of  a  barrel  apiece  ;  how 
many  men  did  he  give  it  to  ? 

15.  How  many  halves  are  there  in  2  ?  * 

16.  In  3  bushels  of  corn  how  many  half  bushels  ? 

17.  How  many  halves  are  there  in  3  ? 

18.  A  boy  divided  4  oranges  among  his  companions, 
giving  them  1  half  of  an  orange  apiece  ;  how  many 
boys  did  he  give  them  to  ? 

19.  How  many  halves  are  there  in  4  ? 

20.  A  man  having  some  laborers,  gave  them  1  half 
a  dollar  apiece  ;  it  took  3  dollars  and  1  half  a  dollar 
to  pay  them  ;  how  many  laborers  were  there  ? 

21.  How  many  halves  are  there  in  3  and  1  half? 

22.  How  many  halves  are  there  in  5  ? 

23.  How  many  halves  are  there  in  7  and  1  half? 

24.  How  can  you  tell  how  many  halves  there  are 
in  any  number  ? 

Answer.  Since  there  are  2  halves  in  one,  there  must 
be  twice  as  many  halves  as  there  are  whole  ones. 

25.  If  you    had    1    orange,  and   should    divide   it 
among  your  companions,  giving  them  1  third  apiece, 
how  many  could  you  give  it  to  ? 

26.  How  many  thirds  are  there  in  1  ? 

27.  If  you  cut  2  oranges  each  hi  to  3  pieces,  how 
many  pieces  would  they  make  ? 

28.  If  you  cut  3  oranges  into  3  pieces  each,  how 
many  pieces  would  they  make  ? 

29.  If  you  cut  4  apples  each   into  3  pieces,  how 
many  pieces  would  they  make  ? 

30.  How  many  thirds  are  there  in2?in3?in4? 
in5? 

31.  How  can  you  tell  how  many  thirds  there  are  io 
any  number  ? 

Answer.  Since  there  are  3  thirds  in  one,  there  will 
be  3  times  as  many  thirds  as  there  are  whole  ones  ? 
*  See  tfao  Key 


Sect.  8.]  ARITHMETIC.  9} 

32.  If  you  had  2  bushels  and  1  third  of  a  bushel 
of  corn  to  give  to  some  poor  persons,  how  many  could 
you  give  it  to  if  you  should  give  them  1  third  of 
a  bushel  apiece  ? 

33.  How  many  thirds  are  there  in  2  and  1  third  ? 

34.  If  a  horse  can  eat  1  third  of  a  bushel  of  oats 
in  1  day,  how  many  days  would  it  take  him  to  eat 
3  bushels  and  2  thirds  of  a  bushel  ? 

35.  How  many  thirds  are  there  in  3  and  2  thirds  ? 

36.  If  1  horse  can  eat  1  third  of  a  bushel  of  oats 
in  a  day,  how  many  horses  will  it  take  to  eat  5  bushels 
and  2  thirds  of  a  bushel  in  the  same  time  ? 

37.  In  5  and  2  thirds  how  many  thirds  ? 

38.  In  7  and  1  third  how  many  thirds  ? 

39.  If  1  horse  will  eat  1  fourth  of  a  ton  of  hay  in 
1  month,  how  many  horses  will  eat  a  ton  in  the  same 
time  ?    How  many  will  eat  2  tons  ?    How  many  will 
eat  3  tons  ?  4  tons  ?  5  tons  ?  6  tons  ? 

40.  How  many  fourths  are  there  inl?  in2?  in3? 
in  4  ?  in  5  ?  in  6  ? 

41.  How  can  you  tell  how  many  fourths  there  are 
in  any  number  ? 

42.  How  many  fourths  are  there  in  4  and  1  fourth  ? 

43.  In  5  tons  of  hay  and  3  fourths  of  a  ton  how 
many  fourths  of  a  ton  ? 

44.  In  5  and  3  fourths  how  many  fourths  ? 

45.  In  7  and  1  fourth  how  many  fourths  ? 

46.  In  9  **nd  3  fourths  how  many  fourths  ? 

47.  If  a  horse   eat  1  fifth  of  a   ton   of  hay  in   a 
month,  how  many  horses  will  eat  a  ton  in  the  same 
time  ?  2  tons  ?  3  tons  ?  4  tons  ?  5  tons  ? 

48.  How  many  fifths  are  there  in  1?  in2?  in  3? 
in  4  ?  in  5  ?  in  7  ?  in  9  ? 

49.  How  can  you  tell  how  many  fifths  there  are  in 
any  number  ? 

50.  In  2  and  1  fifth  how  many  fifths  ? 

51.  In  3  dollars  and  2  fifths  of  a  dollar  how  many 
fifths  of  a  dollar  ? 


92  ARITHMETIC.  [Part  1 

52.  Jn  3  and  2  fifths  how  many  fifths  ? 

53.  In  5  and  3  fifths  how  many  fifths  ? 
•  54.  In  6  and  4  fifths  how  many  fifths  ? 

55.  How  many  sixths  are  there  in  1  ?  in  2  ?  in  3  ? 
in  4  ?  in  5  ?  in  7  ?  in  8  ? 

56.  In  2  and  2  sixths  how  many  sixths  ? 

57.  In  3  and  4  sixths  how  many  sixths  ? 

58.  In  6  and  5  sixths  how  many  sixths  ? 

59.  How  many  sevenths  are  there  in  1  ?  in  2  ?  in 
3?  in  4?  in  6?  "in  9?  in  10? 

60.  In  3  and  1  seventh  how  many  sevenths? 

61.  In  5  and  3  sevenths  how  many  sevenths? 

62.  In  7  and  5  sevenths  how  many  sevenths  ? 

63.  How  many  eighths  are  there  in  1  ?  in  3  ?  in  5  ? 
in  8? 

64.  In  2  and  3  eighths  how  many  eighths  ? 

65.  In  3  and  5  eighths  how  many  eighths  ? 

66.  In  5  and  7  eighths  how  many  eighths  ? 

67.  How  many  ninths  are  there  inl?in2?in7? 
in  5? 

68.  In  2  and  2  ninths  how  many  ninths  ? 

69.  In  4  and  3  ninths  how  many  ninths  ? 

70.  In  6  and  4  ninths  how  many  ninths  ? 

71.  In  8  and  7  ninths  how  many  ninths  ? 

72.  How  many  tenths  in  1  ?  in  2  ?  in  5  ?  in  8  ? 

73.  In  3  and  3  tenths  how  many  tenths  ? 

74.  In  4  and  7  tenths  how  many  tenths  ? 

75.  In  8  and  9  tentlis  how  many  tenths  ? 

76.  In  7  and  4  tenths  how  many  tenths  ? 

77.  In  9  and  8  tenths  how  many  tenths  ? 

78.  In  7  and  4  sevenths  how  many  sevenths  ? 

79.  In  9  and  2  thirds  how  many  thirds  ? 

80.  In  10  and  3  fourths  how  many  fourths  ? 

81.  In  8  and  4  fifths  how  many  fifths  ? 

82.  In  7  and  5  ninths  how  many  ninths  ? 

B.   1.  If  you  give  4  men  1  half  of  a  barrel  of  floui 
apiece,  how  many  barrels  will  it  take  ? 


Sect.  8  ]  ARITHMETIC.  93 

2.  In  4  halves  how  many  times  1  ? 

3.  If  you  give  3  boys  1  half  of  an  orange  apiece, 
how  many  oranges  will  it  take  ? 

t.  In  3  halves  how  many  times  1  ? 
.  If  you  give  five  men  1  half  of  a  dollar  apiece, 
how  many  dollars  will  it  take  ? 

6.  In  5  halves  how  many  times  1  ? 

7.  In  6  halves  how  many  times  1  ? 

8.  In  7  halves  how  many  times  1  ? 

9.  How  can  you  tell  how  many  whole  ones  there  are 
in  any  number  of  halves  ? 

10.  A  man  divided  some  corn  among   6    persons, 
giving  them  1  third  of  a  bushel  apiece ;  how  many 
bushels  did  it  take  ? 

11.  In  6  thirds  how  many  times  1  ? 

12.  In  5  thirds  how  many  times  1  ? 

13.  A  man  gave  eight  paupers  1  third  of  a  dollar 
apiece,  how  many  dollars  did  it  take  ? 

14.  In  8  thirds  how  many  times  1  ? 

15.  In  10  thirds  how  many  times  1  ? 

1 6.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  thirds  ? 

17.  If  a  man  spends  1  fourth  of  a  dollar  in  one  day, 
how  many  dollars  will   he  spend   in  8   days  ?      How 
many  in  7  days  ?  How  many  in  11  days  ? 

18.  In  8  fourths  how  many  times  1  ? 

19.  In  7  fourths  how  many  times  1  ? 

20.  In  1 1  fourths  how  many  times  1  ? 

21.  In  13  fourths  how  many  times  1  ? 

22.  In  18  fourths  how  many  times  1  ? 

23.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  fourths  ? 

24.  If  1  fifth  of  a  barrel  of  flour  will  last  a  family 
1   day,  how  many  barrels  will    last  them    10   days  ? 
How  many  8  days  ?  11  days  ?  17  days  ? 

25.  In  10  fifths  how  many  times  1  ? 

26.  In  8  fifths  how  many  times  1  ? 


94  ARITHMETIC.  [Part  I 

27.  In  1 1  fifths  how  many  times  1  ? 

28.  In  17  fifths  how  many  times  1  ? 

29.  In  18  sixths  how  many  times  1  ? 
80.  In  *2o  fifths  how  many  times  1  ? 

31.  In  21  sevenths  how  many  times  1  ? 

32.  In  24  eighths  how  many  times  1  ? 

33.  In  36  ninths  how  many  times  1  ? 

34.  In  30  tenths  how  many  times  1  ? 

35.  In  35  fourths  how  many  times  1  ? 

36.  In  37  eighths  how  many  times  1  ? 

37.  In  43  fifths  how  many  times  1  ? 

38.  In  48  ninths  how  many  times  1  ? 

39.  In  53  tenths  how  many  times  1  ? 

40.  In  57  eighths  how  many  times  1  ? 

41.  In  76  tenths  how  many  times  1  ? 

42.  In  78  ninths  how  many  times  1  ? 


SECTION  IX. 

A.  1.  IF  a  hreakfast  for  1  man  cost  1  third  of  a 
dollar,  what  would  a  breakfast  for  2  men  cost  ? 

2.  How  much  is  2  times  1  third  ? 

3.  If  it  take  you  1  third  of  an  hour  to  travel  1  mile, 
how  long  will  it  take  you  to  travel  3  miles  ? 

4.  'How  much  is  3  times  1  third  ? 

5.  If  1  man  can  eat  1  third  of  a  pound  of  meat  at 
a  meal,  how  much  can  5  men  eat  ? 

6.  How  much  is  7  times  1  third  ? 

7.  If  1  man  can  eat  2  thirds  of  a  pound  of  meat  for 
dinner,  how  many  thirds  of  a  pound  would  3  men  eat  ? 

8.  How  much  is  2  times  2  thirds  ? 

9.  A  man  gave  to  4  paupers  2  thirds  of  a  dollar 
•ipiece,  how  many  thirds  of  a  dollar  did  he  give  them  ? 
how  many  dollars  ? 

10.  5  times  2  thirds  are  how  many  thirds?  how 
many  times  1  ? 


Sect.  9.]  ARITHMETIC.  95 

1 1 .  Tf  you  gh  e  3  men  1  fourth  of  a  dollar  apiece, 
how  many  fourths  of  a  dollar  will  it  take  ? 

1 2.  3  times  1  fourth  are  how  many  fourths  ? 

13.  If  you  give  3  men  3  fourths  of  a  bushel  of  corn 
apiece,  how  many  fourths  of  a  bushel  will  it  take  ? 
how  many  bushels  ? 

14.  5  times  3  fourths  are  how  many  fourths?  how 
many  times  1  ? 

15.  If  1  horse  eat  1  fifth  of  a  bushel  of  oats  in  a 
day,  how  much  will  4  horses  eat  in  the  same  time  ? 

16.  3  times  1  fifth  are  how  many  fifths  ? 

17.  If  1  man  can  earn  3  fifths  of  a  dollar  in  a  day, 
how  much  can  he  earn  in  4  days  ? 

18.  7  times  3  fifths  are  how  many  fifths  ?  how  many 
times  1  ? 

19.  If  a  family  consume  2  sevenths  of  a  barrel  of 
dour  in  a  week,  how  much  would  they  consume  in  5 
weeks  ? 

20.  6  times   2  sevenths  are  how  many  sevenths  ? 
how  many  times  1  ? 

21.  5  times  3  eighths  are  how  many  eighths  ?  low 
many  times  1  ? 

22.  How  much  is  6  times  3  fifths  ? 

23.  How  much  is  7  times  5  sixths  ? 

24.  How  much  is  5  times  4  ninths  ? 

25.  How  much  is  6  times  8  ninths  ? 

26.  How  much  is  7  times  9  tenths  ? 

27.  How  much  is  5  times  7  tenths  ? 

28.  How  much  is  6  times  7  eighths  ? 

29.  How  much  is  9  times  5  eighths  ? 

30.  How  much  is  8  times  5  sevenths  ? 

31.  How  much  is  7  times  5  sixths  ? 

32.  How  much  is  8  times  7  fourths  ? 

33.  How  much  is  7  times  4  fifths  ? 

34.  How  much  is  5  times  3  eighths  ? 

B.  1.  If  1  bushel  of  wheat  cost  a  dolla,  and  1  half 
what  will  2  bushels  cast  ? 


96  ARITHMETIC.  [Part  I 

2.  How  much  is  2  times  1  and  1  half*  * 

3.  If  a  barrel  of  cider  cost  2  dollars  and  a  half, 
what  will  3  barrels  cost  ? 

4.  How  much  is  4  times  2  and  1  half? 

5.  If  a  barrel  of  beer  cost   3  dollars  and  a  half, 
irhat  will  2  barrels  cost  ? 

6.  How  much  is  5  times  3  and  1  half? 

7.  How  much  is  6  times  3  and  1  half? 

8.  If  a  box  of  butter  cost  2  dollars  and  1  third  of 
a  dollar,  what  will  3  boxes  cost  ? 

9.  How  much  is  4  times  2  and  1  third  ? 

10.  If  you  give  to  two  persons  3  bushels  and  a 
third  of  a  bushel  of  wheat  apiece,  how  many  bushels 
will  it  take  ? 

11.  How  much  is  5  times  3  and  1  third? 

12.  If  you  give  to  4  persons  each  2  oranges  and 
1  fourth    of  an   orange,  how  many  oranges   will  it 
take? 

13.  How  much  is  5  times  2  and  I  fourth  ? 

14.  If  it  take  3  yards  and  2  thirds  of  a  yard  of 
cloth  to  make  a  suit  of  clothes,  how  many  yards  will 
it  take  to  make  2  suits  ? 

15.  How  much  is  4  times  3  and  2  thirds  ? 

16.  If  a  family  consume  2  bushels  and  2  thirds  of  a 
bushel  of  malt  in  1  month,  how  much  will  they  con- 
sume in  3  months  ? 

17.  How  much  is  5  times  2  and  2  thirds  ? 

18.  How  much  is  4  times  3  and  3  fourths  ? 

19.  How  much  is  2  times  3  and  I  fourth  ? 

20.  How  much  is  3  times  3  and  3  fourths  ? 

21.  How  much  is  3  times  5  and  I  fourth  ? 

22.  If  a  horse  eat  3  tons  and  1  fifth  of  a  ton  of 
hay  in  a  year,  how  much  will  2  horses  eat  in  the  same 
time  ? 

23.  How  much  is  4  times  3  and  1  fifth.  ? 

*  This  is  to  be  understood  2  times  1  and  2  times  1  half,  and  tc 
be  answered  thus:  2  times  1  are  2,  and  2  times  1  half  are  2  halves 
or  1,  which,  added  to  2,  rm.kes  3. 


Sect.  9.J  ARITHMETIC.  97 

24.  If  a  man  can  travel  4  miles  and  2  fifths  of  a 
mile  in  one  hour,  how  far  will  he  travel  in  3  hours  ? 

25.  How  much  is  5  times  4  and  2  fifths  ? 

26.  How  much  is  3  times  5  and  §  fifths  ? 

27.  How  much  is  4  times  6  and  2  fifths  ? 

28.  How  much  is  3  times  2  and  1  sixth  ? 

29.  How  much  is  2  times  5  and  3  sixths  ? 

30.  If  a  yard  of  cloth  cost  4  dollars  and  5  sixths  of 
a  dollar,  what  will  4  yards  cost  ? 

31.  How  much  is  7  times  4  and  5  sixths? 

32.  How  much  is  2  times  3  and  3  sevenths  ? 

33.  How  much  is  3  times  4  and  3  sevenths  ? 

34.  If  a  bushel  of  wheat   cost  7  shillings  and  8 
eighths,  what  will  5  bushels  cost  at  that  rate  ?  * 

35.  How  much  is  3  times  4  and  5  eighths  ? 
36-  How  much  is  4  times  3  and  7  eighths  f 

37.  A  man  bought  8  yards  of  cloth,  at  9  dollars  and 
3  tenths  a  yard  ;  how  much  did  it  come  to  ? 

38.  How  much  is  6  times  2  and  2  ninths  ? 

39.  How  much  is  4  times  5  and  3  ninths  ? 

40.  A  man  bought  10  barrels  of  cider  at  3  dollars 
and  5  sixths  a  barrel ;  how  much  did  it  come  to? 

41.  How  much  is  3  times  7  and  2  tenths  ? 

42.  What  cost  8  barrels  of  beef  at  9  dollars  and 
3  sevenths  a  barrel. 

C.  1.  A  boy  wished  to  give  5  other  boys  1  half  pint 
of  chestnuts  apiece  ;  how  many  pints  would  it  take  ? 

2.  A  boy  wished  to  give  3  other  boys  3  fourths  of 
an  orange  apiece  ;  how  many  oranges  would  it  take  ? 

3.  A  man  gave  to  10  persons  1  fifth  of  a  bushel  of 
eorn  apiece  ;  how  many  bushels  did  it  take  ? 

4.  A  man  gave  to  7  men  3  fourths  of  a  gallon  of 
beer  apiece ;  how  many  gallons  of  beer  did  it  take  ? 

5.  If  it  take  1  yard  and  1  fourth  of  a  yard  of  cloth 
to  make  a  pair  of  pantaloons,  how  many  yards  would 
it  take  to  make  8  pairs  ? 


98  ARITHMETIC.  [Part  I. 

6.  If  a  family  consume  2  bushels  and  3  fifths  of  a 
bushel  of  grain  in  1  week,  how  many  bushels  would 
they  consume  in  4  weeks? 

7.  If  a  horse  eat  3  bushels  and  4  sevenths  of  a 
bushel  of  oats  in  1  week,  how  many  bushels  would 
he  eat  in  8  weeks? 

8.  If  a  horse  eat  5  loads  and  3  eighths  of  a  load 
of  hay  in  1  year,  how  many  loads  would  6  horses 
eat? 

9.  If  a  man  travel  4  miles  and  5.  ninths  in  an  hour, 
how  many,  miles  would  he  travel  in  8  hours  ? 

10.  If  in  an  orchard  of  10  trees,  each  tree  bears  8 
bushels  and  3  sevenths,  how  many  bushels  will  the 
whole  orchard  bear? 

11.  If  a  man  can  build  5  rods  and  7  eighths  of  a 
rod  of  wall  in  1  day,  how  many  rods  can  he  build  in 
8  days  ? 

12.  If  3  men  can  build  a  piece  of  wall  in  4  days  and 

3  fifths  of  a  day,  how  many  days  would  it  take  1  man 
to  build  it  ? 

13.  If  1  man  can  build  7  rods  and  2  fifths  of  wall 
in  a  day,  how  many  rods  would  1 0  men  build  ? 

•'  14.  If  1  man  can  build  3  rods  and  2  ninths  of 
wall  in  1  day,  how  many  rods  would  3  men  build 
in  4  days  ? 

15.  If  it  take  1  yard  and  3  sevenths  of  a  yard  of 
cloth  to  make  1  pair  of  pantaloons,  and  2  yards  and 

4  sevenths  for  a  coat ;  how  many  yards  would  it  take 
to  make  3  pairs  of  pantaloons  and  3  coats. 


SECTION  X. 

A.  1.  IF  a  yard  of  cloth  cost  3  dollars,  what  wil) 
1  half  of  a  yard  cost  ? 
2.  What  is  1  half  of  3  ?  * 

*  See  Section  V..  article  A 


Sect.  10.]  ARITHMETIC.  99 

3.  If  a  barrel  of  beer  cost  5  dollars,  what  will  1 
half  of  a  barrel  cost  ? 

4.  What  is  1  half  of  5  ? 

5.  If  2  barrels  of  cider  cost  7  dollars,  what  is  that 
a  barrel  ? 

6.  What  is  1  half  of  7  ? 

7.  What  is  1  half  of  9  ? 

8.  What  is  1  half  of  11  ? 

9.  What  is  1  half  of  13  ? 

10.  What  is  1  half  of  15  ? 

11.  If  you  divide  1  bushel  of  wheat  equally  among 
3  persons,  what  part  of  a  bushel  will  you  give  them 
apiece  ? 

12.  If  .3  yards  of  cloth  cost  1  dollar,  what  part  of 
a  dollar  will  1  yard  cost  ? 

13.  What  is  1  third  of  1  ? 

14.  How  could  you  divide  2  oranges  into  3  equal 
parts  ?  that  is,  how  can  you   find    1  third   of  2  or- 
anges ?  * 

15.  One  third  of  2  oranges  will  be  the  same  as 
how  many  thirds  of  1  orange  ? 

16.  If  you   divide    2    bushels    of  wheat    equally 
among  3  persons,  what  part  of  a  bushel  will  you  give 
them  apiece  ? 

17.  If  3  bushels  of  corn  cost  2  dollars,  what  part 
of  a  dollar  will  1  bushel  cost  ? 

NOTE.  One  third  of  two  things  is  twice  as  much  as 
one  third  of  one  thing.  One  third  of  one  is  one  third, 
and  consequently  one  third  of  two  things  is  two  thirds. 
In  the  same  manner,  one  third  of  four  things  is  four 
thirds  of  one  thing.  If  four  oranges  be  cut  each  into 
three  parts,  and  then  one  part  of  each  be  taken,  it  will 
make  four  pieces,  each  of  which  is  one  third  of  one 

*  Divide  each  orange  into  three  parts,  and  then  take  one  part 
from  each. 


100  AU1THMET1C.  [Part  1 

orange.     Hence  one  third  of  four  oranges  is  four  third* 
of  one  orange,  that  is,  one  whole  one  and  one  third. 

18.  If  3  bushels  of  wheat  cost  4  dollars,  how  much 
is  that  a  bushel  ? 

19.  What  is  one  third  of  2  ?  of  4  ? 

20.  If  3  gallons  of  wine  cost  5  dollars,  what  is  that 
a  gallon  ? 

21.  What  is  1  third  of  5  ?  of  7  ?  of  8  ?  of  10  ?  of 
11. 

22.  If  a  bushel  of  apples  be  divided  equally  among 

4  persons,  what  part  of  a  bushel  will  they  have  apiece  ? 
What  would  they  have  apiece,  if  2  bushels  were  di- 
vided among  them  ?    What  if  3  bushels  ?    What  if  5 
bushels  ?  What  if  6  bushels  ? 

23.  What  is  1  fourth  of  1  ?  of  2  ?  of  3  ?  of  5  ?  of 
6  ?  of  7  ?  of  9  ?  of  10  ? 

24.  If  a  bushel  of  malt  will  serve  5  persons  1  month, 
how  much  will  serve  1  person  the  same  time  ? 

25.  If  2  barrels  of  cider  will  serve  5    persons   1 
month,  how  much  will  serve  1  person  the  same  time. 

26.  If  3  barrels  of  flour  be  divided  among  5  men, 
how  much  will  each  have  ?  If  4  barrels  were  divided, 
what  would    each   have  ?     What   if  6    barrels  were 
divided  ?  What  if  7  barrels  were  divided  ? 

27.  What  is  1  fifth  of  1  ?  of  2  ?  of  3  ?  of  4  ?  of 
6  ?  of  7  ? 

28.  What  is  1  sixth  of  1  ?  of  2  ?  of  3  ?  of  4  ?  of 

5  ?  of  7  ?  of  8  ?  of  9  ?  of  10  ? 

29.  What  is  1  seventh  of  1  ?  of  2  ?  of  3  ?  of  4  ? 
of  5  ?  of  6  ? 

30.  What  is  1  eighth  of  1  ?  of  2  ?  of  3  ?  of  4  ? 
of  5  ?  of  6  ?  of  7  ?  of  8  ?  of  9  ?.  of  10  ? 

31.  What  is  1  ninth  of  1  ?   of  2  ?  of  3  ?   of  4  ? 
of  5  ?  of  6  ?  of  7  ?  of  8  ?  of  9  ?  of  10  ?  of  11  ? 

32.  What  is  1  tenth  of  1  ?   of  2  ?    of  3  ?   of  4  ? 
of  5  ?  of  6  ?  of  7  ?  of  8  ?  of  9  ?  of  10  ?  of  11  ?  of 
12?  of  13? 


Sect.  10.]  ARITHMETIC.  101 

33.  If  3  yards  of  cloth  cost  2  dollars,  what  will  1 
yard  cost  ?  What  will  2  yards  cost  ?     I    ;  ,.  > 

34.  If  1  bushel  of  wheat  cost  2  dollpir.VVhaVwill 
1  third  of  a  bushel  cost  ?     What  will  2  thirds  of  a 
bushel  cost  ? 

35.  What  is  1  third  of  2  ? 

36.  What  is  2  thirds  of  2  ? 

37.  If  a  load  of  wood  cost  5  dollars,  what  will  1 
third  of  a  load  cost  ?     What  will  2  thirds  of  a  load 
cost? 

38.  What  is  1  third  of  5  ? 

39.  What  is  2  thirds  of  five  ? 

40.  What  is  2  thirds  of  7  ? 

41.  If  4  bushels  of  salt  cost  3  dollars,  what  will  1 
bushel  cost  ?  WTiat  will  3  bushels  cost  ? 

42.  What  is  1  fourth  of  3  ? 

43.  What  is  3  fourths  of  3  ? 

44.  If  a  barrel  of  cider  cost  2  dollars,  what  will  1 
fifth  of  a  barrel  cost  ?    What  will  2  fifths  of  a  barrel 
cost? 

45.  What  is  1  fifth  of  two  ? 

46.  What  is  2  fifths  of  2  ? 

47.  What  is  1  fifth  of  7  ? 

48.  What  is  3  fifths  of  7  ? 

49.  If  7  gallons  of  wine  cost"  5  dollars,  what  will  a 
gallon  cost  ?  What  will  4  gallons  cost  ? 

50.  What  is  1  seventh  of  5  ? 

51.  What  is  4  sevenths  of  5  ? 

52.  What  is  1  sixth  of  4  ? 

53.  What  is  5  sixths  of  4  ? 

54.  If  you  divide  7  dollars  among  8  men,  what  part 
of  a  dollar  will  you  give  them  apiece  ?    What  would 
three  of  them  have  ? 

55.  What  is  1  eighth  of  7  ? 

56.  What  is  3  eighths  of  7  ? 

57.  What  is  1  ninth  of  10  ? 

58.  What  is  7  ninths  of  10  ? 


102  ARITHMETIC.  [Part  i. 

59.  What  is  1  tenth  of  14  ? 

60.  What  is  '4  tenths  of  14  ? 

61.  If  5  yar<Is  of  cloth  cost  17  dollars,  what  is  that 
a  ya*-f]  ?  What  would  3  yards  cost  ?   What  would  8 
Cards' cost  ?-  •-• 

62.'  What  is  3  fifths  of  17  ? 

63.  What  is  8  fifths  of  17  ? 

64.  What  is  5  sevenths  of  20  ? 

65.  What  is  2  ninths  of  22  ? 

66.  What  is  3  eighths  of  27  ? 

67.  What  is  2  thirds  of  28  ? 

68.  What  is  3  fourths  of  31  ? 

69.  If  5  loads  of  hay  cost  47  dollars,  what  is  that 
a  load  ?   What  will  2  loads  cost  ?   What  will  7  loads 
cost  ?  What  will  12  loads  cost  ? 

70.  What  is  2  fifths  of  47  ? 

71.  What  is  7  fifths  of  47  ? 

72.  What  is  12  fifths  of  47  ? 

73.  What  is  4  sevenths  of  48  ? 

74.  What  is  4  ninths  of  50  ? 

75.  What  is  2  sevenths  of  58  ? 

76.  What  is  3  eighths  of  61  ? 

77.  What  is  4  tenths  of  73  ? 

78.  What  is  8  ninths  of  65  ? 

79.  What  is  9  tenths  of  78  ? 

80.  What  is  7  eighths  of  70  ?    . 

B.  1.  If  you  divide  7  apples  equally,  between   2 
boys,  how  many  would  you  give  them  apiece  ? 

2.  How  can  you  divide  5  oranges  equally  among 
3  persons  ? 

3.  Divide  3  bushels  of  corn  equally  among  5  men, 
how  much  would  you  give  them  apiece  ? 

4.  A  boy  had  7  pears,  and  gave  away  3  fifths  of 
them ;  how  many  did  he  give  away  ?  and  how  many 
had  he  left  ? 

5.  If  2  yards  of  cloth  cost  3  dollars,  what  is  that  a 
yard  ? 


Sect.  11.]  ARITHMETIC.  103 

6.  If  4  yards  of  cloth  cost  2  dollars,  what  is  that  a 
yard? 

[Let  the  answers  be  given  in  dollars  and  cents  or  in 
shillings.     1  dollar  is  100  cents.] 

7.  If  5  bushels  of  corn  cost  7  dollars,  what  is  that 
a  bushel  ? 

8.  If  a  man  receive  8-  dollars  for  6  days'  work, 
what  is  that  per  day  ? 

9.  If  3  bushels  of  wheat  cost  8  dollars,  what  wiD 

2  bushels  cost  ? 

10.  A  man  had  30  dollars,  and  gave  away  3  sev- 
enths of  it ;  how  much  did  he  give  away  ? 

11.  If  4  yards  of  cloth  cost  10  dollars,  what  will 

3  yards  cost  ? 

12.  If  3  barrels  of  cider  cost  8  dollars,  what  will 
10  barrels  cost  ? 

13.  If  7  pounds  of  flour  cost  40  cents,  what  will 
10  pounds  cost  ? 

14.  If  4  firkins  of  butter  cost  26  dollars,  what  will 
7  firkins  cost  ? 

15.  If  3  men  can  do  a  piece  of  work  in  7  days, 
how  long  will   it  take   1  man  to  do  it-?    How  long 
would  it  take  4  men  ? 

16.  If  2  pipes  will  empty  a  cistern  in  3  hours,  in 
how  long  a  time  would  1  empty  it  ?    In  how  long  a 
time  would  7  pipes  empty  it  ? 


SECTION  XI. 

A.  1.  BOUGHT  1  half  of  a  yard  of  cloth  for  1  dol- 
lar and  1  half ;  what  was  that  a  yard  ? 

2.  1  and  1  half  is  the  half  of  what  number  ?  * 

3.  If  1  third  of  a  yard  of  cloth  cost  1  dollar  and  3 
fourths  of  a  dollar,  how  much  does  a  yard  cost  ? 

*  See  Sect.  VI.  art.  A,  and  Sect.  IX.  art.  B. 


104  ARITHMETIC.  [Part  1 

4.  2  and  3  fourths  is  1  third  of  what  number  ? 

5.  If  1  half  of  a  barrel  of  beer  cost  2  dollars  and 
1  fourth  of  a  dollar,  how  much,  will  a  barrel  cost  ? 

6.  2  and  3  fourths  is  one  half  of  what  number  ? 

7.  If  1  fourth  of  a  box  of  lemons  cost   3   dollars 
and  2  thirds  of  a  dollar,  what  will  a  box  cost  ? 

8.  2  and  2  thirds  is  1  fourth  of.  what  number  ? 

9.  3  and  1  half  is  1  fourth  of  what  number  ? 

10.  If  1  third  of  a  barrel  of  pork  cost  4  dollars 
and  1  third  of  a  dollar,  what  will  a  barrel  cost  ? 

11.  4  and  2  thirds  is  1  third  of  what  number? 

12.  If  1  sixth  of  a  barrel  of  fish  cost  2  dollars  and 
1  fifth  of  a  dollar,  what  will  a  barrel  cost  ? 

13.  3  and  2  fifths  is  1  sixth  of  what  number  ? 

14.  If  1  fifth  of  a~  barrel  of  salmon  cost  3  dollars 
and  2  sevenths  of  a  dollar,  what  is  that  a  barrel  ? 

15.  4  and  3  sevenths  is  1  fifth  of  what  number  ? 

16.  If  a  man  can  travel  4  miles  and  3  sevenths  of 
a  mile  in  1  ninth  of  a  day,  how  far  will  he  travel  in  a 
whole  day  ? 

17.  5  and  3  sevenths  is  1  ninth  of  what  number  ? 

18.  2  and  4  fifths  is  1  seventh  of  what  number  ? 

19.  6  and  3  eighths  is  1  eighth  of  what  number  ? 

20.  7  and  5  sixths  is  1  tljird  of  what  number  ? 

21.  8  and  3  sevenths  is  1  fourth  of  what  number  ? 

22.  5  and  3  tenths  is  1  seventh  of  what  number  ? 

23.  8  and  5  ninths  is  1  fifth  of  what  number  ? 

24.  9  and  4  fifths  is  1  eighth  of  what  number  ? 

25.  6  and  4  tenths  is  1  tenth  of  what  number  ? 

26.  7  and  4  ninths  is  1  fifth  of  what  number  ? 

27.  8  and  7  tenths  is  1  seventh  of  what  number  ? 

B.  1.  If  2  thirds  of  a  barrel  of  beef  cost  3  dollars, 
what  does  1  third  of  a  barrel  cost  ? 
2.  3  is  2  times  what  number  ?  * 
Ans.  3  is  2  times  the  half  of  3  ;  but  1  half  of  3  is  3 
*  See  Section  VI.,  article  B. 


Sect.  11.]  ARITHMETIC.  105 

halves,  or  1  and  1  half,*  therefore  3  is  2  times  1  and 
1  half. 

3.  If  3  fifths  of  a  yard  of  cloth  cost  4  dollars,  what 
will  1  fifth  of  a  yard  cost  ? 

4.  5  is  3  times  what  number  ? 

5.  If  3  sevenths  of  a  barrel  of  pork  cost  2  dollars, 
what  will  1  seventh  of  a  barrel  cost  ? 

6.  2  is  3  times  what  number  ? 

7.  If  4  thirds  of  a  bunch  of  shingles  cost  5  dollars, 
what  does  1  third  of  a  bunch  cost  ? 

8.  7  is  4  times  what  number  ? 

9.  2  is  4  times  what  number  ? 

10.  A  man  bought  4  ninths  of  a  barrel  of  flour  for 
3  dollars ;  what  would  be  the  price  of  1  ninth  of  a 
barrel  at  the  same  rate  ? 

11.  5  is  4  times  what  number  ? 

12.  A  man  bought  5  eighths  of  a  hundred-weight 
of  sugar  for  6  dollars  ;  what  would  1  eighth  of  a  hun- 
dred-weight cost  at  the  same  rate  ? 

13.  7  is  5  times  what  number? 

14.  3  is  5  times  what  number  ? 

15.  8  is  5  times  what  number  ? 

16.  9  is  4  times  what  number  ? 

17.  11  is  6  times  what  number  ? 

18.  13  is  7  times  what  number  ? 

19.  14  is  8  times  what  number  ? 

20.  17  is  5  times  what  number  ? 

21.  18  is  8  times  what  number  ? 

22.  17  is  9  times  what  number  ? 

23.  15  is  10  times  what  number? 

24.  20  is  9  times  what  number  ? 

25.  22  is  10  times  what  number  ? 

26.  24  is  7  times  what  number? 

27.  If  2  thirds  of  a  barrel  of  beef  cost  3  dollars, 
what  will  1  third  of  a  barrel  cost  ?     What  will  the 
whole  barrel  cost  ? 

*  See  Section  X.,  article  A. 


106  ARITHMETIC.  [Part  \. 

28.  *  If  5  is  2  thirds  of  some  number,  what  is  1 
third  of  the  same  number  ?  2  and  1  half  is  1  third  of 
what  number  ?    Then  5  is  two  thirds  of  what  number  r 

29.  If  3  fourths  of  a  barrel  of  flour  cost  5  dollars, 
what  will  1  fourth  of  a  barrel  cost  ?  What  will  the 
whole  barrel  cost  ? 

30.  If  8  is  3  fourths  of  some  number,  what  is  1 
fourth  of  the  same  number  ?  2  and  2  thirds  is  1  fourth 
of  what  number  ?  Then  8  is  3  fourths  of  what  num- 
ber ? 

31.  A  man  bought  2  sevenths  of  a  barrel  of  oil  for 
5  dollars  ;  how  much  will  1  seventh  cost  at  the  same 
rate  ?  how  much  would  a  barrel  cost  ? 

32.  If  9  is  2  sevenths  of  some  number,  what  is  1 
seventh  of  the  same  number  ?  4  and  1  half  is  1  sev- 
enth of  what  number  ?  Then  9  is  2  sevenths  of  what 
number  ? 

33.  Bought  5  eighths  of  a  chaldron  of  coal  for  7 
dollars ;  what  is  the  price  of  1  eighth,  at  the  same 
rate  ?  What  is  the  price  of  the  whole  chaldron  ? 

34.  If  12  is  5  eighths  of  some  number,  what  is  1 
eighth  of  the  same  number  ?  2  and  2  fifths  is  1  eighth 
of  what  number?  Then  12  is  5  eighths  of  what  num- 
ber? 

35.  If  a  man  can  do  7  ninths  of  a  piece  of  work  in 
4  days,  how  long  would  it  take  him  to  do  1  ninth  of 
it  ?  how  long  would  it  take  him  to  do  the  whole  ? 

36.  If  3  is  7  ninths  of  some  number,  what  is   1 
ninth  of  the  same  number  ?  3  sevenths  is  1  ninth  of 
what  number  ?  Then  3  is  7  ninths  of  what  number  ? 

37.  If  3  fifths  of  a  chaldron  of  coal  cost  8  dollars, 
what  is  the  whole  chaldron  worth  ? 

38.  7  is  3  fifths  of  what  number? 

39.  A  man  bought  6  sevenths  of  a  cask  of  raisins 
for  5  dollars  ;  what  was  the  whole  cask  worth  ? 

40.  8  is  6  sevenths  of  what  number  ? 

*  See  Sect.  VI.,  art.  B,  examples  8  and  following. 


Sect.  11.]  ARITHMETIC.  107 

41.  A  man  had  4  sixths  of  a  week's  board  for  S 
dollars  ;  how  much  is  that  for  a  whole  week  ? 

42.  3  is  5  sixths  of  what  number  ? 

43.  9  is  5  eighths  of  what  number  ? 

44.  10  is  7  thirds  of  what  number? 

45.  1 1  is  6  fifths  of  what  number  ? 

46.  12  is  7  fifths  of  what  number  ? 

47.  15  is  2  sevenths  of  what  number  ? 

48.  17  is  3  tenths  of  what  number  ? 

49.  A  man  bought  5  eighths  of  a  cask  of  wine  for 
19  dollars,  what  would  the  whole  cask  cost  ? 

50.  19  is  3  eighths  of  what  number? 

51.  21  is  4  fifths  of  what  number  ? 

52.  Bought  5  ninths  of  a  ton  of  logwood  for  23  dol- 
lars ;  what  would  a  ton  cost  at  that  rate  ? 

53.  23  is  7  ninths  of  what  number  ? 

54.  21  is  3  tenths  of  what  number? 

55.  29  is  4  sevenths  of  what  number  ? 

56.  31  is  5  eighths  of  what  number  ? 

57.  33  is  6  sevenths  of  what  number  ? 

58.  38  is  4  ninths  of  what  number  ? 

59.  A  man  bought  1  barrel  and  1  seventh  of  a  bar- 
rel of  wine  for  41  dollars  ;  what  was  that  a  barrel  ? 

NOTE.  —  1  barrel  and  1  seventh  of  a  barrel  is  the 
same  as  8  sevenths  of  a  barrel.  If  8  sevenths  of  a  bar- 
rel cost  .41  dollars,  what  does  a  barrel  cost  ? 

60.  35  is  8  sevenths  of  what  number  ? 

61.  A  man  bought  1  ton  and  4  fifths  of  a  ton  (that 
is,  9  fifths  of  a-  ton)  of  fustic  for  43  dollars,  what  was 
that  a  ton  ? 

62.  52  is  9  fifths  of  what  number?     - 

63.  Bought  1  ton  and  2  sevenths  of  a  ton  of  log- 
wood for  48  dollars,  what  was  that  a  ton  ? 

64.  67  is  9  sevenths  of  what  number  ? 

65.  53  is  7  tenths  of  what  number  ? 

66.  58  is  9  eighths  of  what  number  ? 

67.  61  is  10  ninths  of  what  number  ? 


108  ARITHMETIC.  [Part  1 

68.  Bought  2  barrels  and  1  fifth  of  a  barrel  of  oil 
(that  is,  11  fifths  of  a  barrel)  for  65  dollars  ;  what 
was  that  a  barrel? 

69.  65  is  9  fifths  of  what  number  ? 

70.  71  is  8  elevenths  of  what  number? 

C.  1.  A  boy  gave  away  2  apples  and  1  half,  which 
was  1  fourth  of  all  he  had ;  how  many  had  he  ?   . 

2.  A  man  gave  away  3  dollars,  which  was  2  fifths 
of  all  the  money  he  had  ;  how  much  had  he  ? 

3.  A  man  sold  a  cow  for  fifteen  dollars,  which  was 
4  fifths  of  what  she  cost  him  ;  how  much  did  he  lose 
by  the  bargain  ? 

4.  A  man  sold  a  piece   of   cloth   for  37  dollars, 
which  was  9  eighths  of  what  it  cost  him ;  how  much 
did  he  gain  by  the  bargain  ? 

5.  There  is  a  pole  3  fifths  under  water,  and  7  feet 
out  of  the  water  ;  how  long  is  the  pole  ? 

6.  A  man  sold  a  piece  of  cloth  for  47  dollars,  by 
which  bargain  he  lost  2  ninths  of  what  the  cloth  cost 
him ;  how  much  did  it  cost  him,  and  how  much  did 
he  lose  ? 

Miscellaneous  Examples. 

1.  If  a  staff  5  feet  long  cast  a  shadow  4  feet  at  12 
o'clock,  what  is  the  length  of  a  pole  that  casts  a^shadow 
67  feet  at  the  same  time  ? 

2.  If  53  gallons  of  water,  in  1  hour,  run  into  a  cis- 
tern containing  97  gallons,  and  44  gallons  run  out  in 
an  hour,  in  what  time  will  it  be  filled  ? 

3.  A  man  bought  a  cask  of  wine  containing  75 
gallons  ;  2  sevenths  of  it  leaked  out,  and  he  sold  the 
remainder  for  1  dollar  a  gallon  ;  how  much  did  he  sell 
it  for  ? 

4.  A  cask  of  wine  cost  67  dollars  ;  what  is  5  eighths 
of  it  worth  ? 

5.  A  man  bought  9  oranges  for  6  cents  and  2  sev- 


Sect.  12.]  ARITHMETIC.  109 

enths  apiece,  and  sold  them  for  67  cents  ;  what  did  he 
gain  by  the  bargain  ? 

6.  A  man  bought  10  yards  of  broadcloth  for  70 
dollars  ;  how  must  he  sell  it  per  yard  in  order  to  gain 
14  dollars  ? 

7.  If  when  the  days  are  12  hours  long,  a  man  per- 
form a  journey  in  3  days,  how  many  hours  is  he  in 
performing  it  ? 

8.  If  a  man  perform  a  journey  in  36  hours,  how 
many  days  would   he  be  in  performing  it  when  the 
days  are  nine  hours  long  ? 

9.  If  when  the  days  are  11  hours  long,  a  man  can 
perform  .a  journey  in  5  days,  in  how  many  hours  will 
he  perform  it  ?  In  how  many  days  when  the  days  are 
9  hours  long  ? 

10.  What  number  added   to  2   fifths  of  33,  will 
make  the  number  17  ? 

11.  How  many  yards  of  cloth,  that  is  1  quarter  of 
a  yard  wide,  will  line  10  yards  that  is  3  quarters  wide  ? 

12.  8  yards  of  cloth,  that  is  1  quarter  wide,  are 
equal  to  how  many  yards  that  is  4  quarters  wide  ? 

13.  How  many  yards  of  cloth,  that  is   3  quarters 
wide,  are  equal  to  7  yards  that  is  5  quarters  wide  ? 

14.  How  many  yards  of  cloth,  that  is  6  quarters 
wide,  are  equal  to  37  that  is  4  quarters  wide  ? 

15.  If  a  piece  of  cloth,  5  quarters  wide,  be  worth 
37  dollars,  what  is  a  piece  oT  the  same  length,  3  quar- 
ters wide,  worth  ? 

16.  If  cloth  4  quarters  wide  is  worth  8  dollars  a 
yard,  what  is  1  yard  of  the  same  kind  of  cloth,  that 
is  5  quarters  wide,  worth  ? 


SECTION  XII. 

PARTS  of  one  are  called  fractions.     Fractions  may 
be  expressed  by  figures,  as  well  as  whole  numbers, 


110  ARITHMETIC.  [Part  1. 

It  requires  two  numbers  to  express  a  fraction  ;  one 
to  show  into  how  many  parts  one  is  divided,  and  the 
other  to  show  how  many  of  those  parts  are  used. 
For  example,  if  we  wish  to  express  one  half,  (which 
means  that  one  is  divided  into  two  equal  parts,  and 
that  one  part  is  used,)  we  must  use  the  figure  2  to 
express  that  one  is  divided  into  two  equal  parts  and 
the  figure  1  to  show  that  one  part  is  used.  And  these 
must  be  written  in  such  a  manner  that  we  may  always 
know  what  each  of  them  is  intended  to  express. 

One  half  is  usually  written  thus,  £ ;  one  number 
above  a  line,  and  the  other  below  it.  The  number 
below  the  line  shows  into  how  many  parts  one  is 
divided,  and  the  number  above  the  line  shows  how 
many  parts  are  used. 

One  third  is  written  £ 

Two  thirds       -  § 

One  fourth       -----£ 
Three  fourths  -  J 

Two  fifths        -  £ 

Example.  %  of  an  apple  signifies  that  the  apple  is 
to  be  cut  into  7  equal  parts,  and  that  3  parts  are  to 
be  used. 

Illustrate  by  a  line,  divided  into  7  equal  parts,  and 
three  of  the  parts  taken.  In  the  same  way  illustrate 
the  meaning  of  the  fractions  f ,  f ,  ^. 

We  may  observe,  that,  when  one  is  divided  into  3 
parts,  the  parts  are  called  thirds  ;  when  one  is  divided 
into  4  parts,  the  parts  are  called  fourths,  &c. ;  that  is, 
the  fraction  takes  its  name  from  the  number  of  parts 
into  which  one  is  divided.  The  number  under  the 
line  is  called  the  denominator,  because  it  gives  name 
to  the  fraction  ;  and  the  number  above  the  line  is 
called  the  numerator,  because  it  shows  the  number  of 
parts  used.  Thus  ^,  10  is  the  denominator,  and  3 
the  numerator. 

N.  B.     The  pupil  must  be  made  familiar  with  this 


beet.  12.}  ARITHMETIC.       ,  111 

mode  of  expressing  fractions,  and  must  be  able  to 
apply  it  to  any  familiar  objects  ;  as  apples,  oranges, 
&c. ;  *  or  by  blackboard,  before  he  is  allowed  to  pro- 
ceed any  farther.  Particular  care  must  be  taken  to 
•make  him  understand  what  the  denominator  signifies, 
and  what  the  numerator,  as  explained  above.  The 
denominator  should  always  be  explained  first. 

The  following  examples  are  a  recapitulation  of 
some  of  the  foregoing  sections,  for  the  purpose  of 
showing  the  application  of  the  above  method  of  writ- 
ing fractions.  Having  analyzed  the  question,  the 
pupil  may  write  the  required  fraction  on  the  black- 
board. 


See  Section  VIII.     A. 

A.  1.  In  2  how  many  times  J  ?  * 

2.  In  3  how  many  times  £  ? 

3.  In  2  how  many  times  |  ? 

4.  In  4  how  many  times  J  ? 

5.  In  6  how  many  times  £  ? 

6.  In  7  how  many  times  £  ? 

7.  In  8  how  many  times  £  ? 

8.  In  2  Jt  how  many  times  J  ? 

9.  In  3^  how  many  times  £  ? 

10.  Reduce  4^  to  an  improper  fraction.  J 

11.  Reduce  3f  to  an  improper  fraction. 
.12.  Reduce  of  to  an  improper  fraction. 

18.  Reduce  6f  to  an  improper  fraction. 

14.  Reduce  8T3o  to  an  improper  fraction. 

15.  Reduce  9  f  to  an  improper  fraction. 

*  When  the  numerator  is  larger  than  the  denominator,  the  frac- 
tion is  called  an  improper  fraction.  See  Key. 

t  2i  is  read  2  and  1  half.     It  is  called  a  mixed  number. 

t  That  is,  to  find  how  many  fifths  there  are  in  4  and  1  fifth 
first,  knd  how  many  fifths  there  are  in  4. 


112  ARITHMETIC.  [Part  I. 

B.  1.  |  are  how  many  times  1  ? 

2.  |  are  how  many  times  1  ? 

3.  J  are  how  many  times  1  ? 

4.  f  are  how  many  times  1  ? 

5.  §  are  how  many  times  1  ? 

6.  Y*  are  h°w  many  times  1  ? 

7.  ^8  are  how  many  times  1  ? 

8.  *?  are  how  many  times  1  ? 

9.  %/  are  how  many  times  1  ? 
10.  f  J  are  how  many  times  1  ? 


/See  Section  IX. 

^ 

A.  1.  How  much  is  3  times  £  ? 

2.  How  much  is  4  times  |  ? 

3.  How  much  is  3  times  £  ? 

4.  How  much  is  4  times  ^  ? 

5.  How  much  is  5  times  |  ? 

6.  How  much  is  6  times  J  ? 

7.  How  much  is  8  times  ^  ? 

8.  How  much  is  9  times  \  ? 

9.  How  much  is  10  times  |  ? 
10.  How  much  is  9  times  §  ? 

B.  1.  How  much  is  3  times  2£  ? 

2.  How  much  is  4  times  3§  ? 

3.  How  much  is  5  times  6^  ?    ^ 

4.  How  much,  is  6  times  4f  ? 

5.  How  much  is  7  times  5f  ? 

6.  How  much  is  8  times  6|  ? 

7.  How  much  is  4  times  10^  ? 

8.  How  much  is  9  times  7 J  ? 

9.  How  much  is  8  times  9f  ? 
10.  How  much  is  10  times  7$? 


Sect.  12.] 


ARITHMETIC. 


lid 


See  Sections  -V.  &  X. 


1. 

3. 
5. 
7. 
9. 
11. 
13. 
15. 
17. 
19. 

What 
What 
What 
What 
What 
What 
What 
What 
What 
What 

is 
is 
is 
is 
is 
is 
is 
is 
is 
is 

j 

i 
§ 

! 
1 

o 

y 
i 

f 
f 
f 

of 
of 
of 

of 
of 
of 
of 
of 
of 
of 

6? 
8? 
9? 
14? 
5? 
7? 
17? 
27? 
47? 
65? 

2. 

4. 
<5. 
8. 
10. 
12. 
14. 
16. 
18. 
20. 

What 
What 
What 
What 
What 
What 
What 
What 
What. 
What 

is 
is 
is 
is 
is 
is 
is 
is 
is 
is 

j 

i 

\ 

i 

3 
4 
f 
f 
t 

i 
? 

of 
of 
of 
of 
of 
of 
of 
of 
of 
of 

6? 
9? 
10? 
5? 
7? 
35? 
26? 
37? 
42? 
75? 

See  Sections  VI. 

'A.  1.  2  is  J  of  what  numher  ? 

2.  4  is  J  of  what  number  ? 

3.  8  is  \  of  what  number  ? 

4.  1 J  is  J  of  what  number  ? 

5.  2f  is  £  of  what  number  ? 

6.  4f  is  J  of  what  number  ? 

7.  6|  is  |  of  what  number  ? 

8.  7f  is  j  of  what  number  ? 

9.  8  j  is  J  of  what  number  ? 
10.  9^  is  ^  of  what  number  ? 

B.  1.  4  is  §  of  what  number  ? 

2.  6  is  f  of  what  number  ? 

3.  8  is  |  of  what  number  ? 

4.  12  is  f  of  what  number? 

5.  15  is  f  of  what  number? 

6.  18  is  f  of  what  number? 

7.  20  is  f  of  what  number? 

8.  24  is  f  of  what  number  ? 

9.  28  is  £  of  what  number  ? 
10.  30  is  ^  of  what  number  ? 

8 


114  ARITHMETIC.  [Part  1. 

11.  3  is  §  of  what  number? 

12.  4  is  §.  of  what  number  ? 

13.  5  is  $  of  what  number? 

14.  8  is  §  of  what  number  ? 

15.  9  is  f  of  whal^number? 

16.  17  is  j  of  what  number? 

17.  25  is  J  of  what  number  ? 

18.  38  is  §  of  what  number? 

19.  43  is  f  of  what  number? 

20.  54  is  y>  of  what  number  ? 

Miscellaneous  Examples. 

1.  A  man  sold  8  yards  of  cloth  for  3f  dollars  a 
yard ;  what  did  it  come  to  ? 

2.  A  man  sold  a  horse  for  76  dollars,  winch  was  § 
of  what  it  cost  him  ;  how  much  did  it  cost  him  ? 

3.  A  man  sold  £  of  a  gallon  of  wine  for  40  cents  ; 
what  was  that  a  gallon  ? 

4.  If  it  will  take  1J  yards  of  cloth  to  make  a  coat, 
how  many  yards  will  it  take  to  make  7  coats  ? 

5.  If  1  horse  consume  3f  bushels  ~of  oats  in  2  days, 
how  much  would  2  horses  consume  in  5  days  ? 

6.  If,  when  the  days  are  9J  hours  long,  a  man  per- 
form a  journey  in  10  days,  in  how  many  days  would 
he  perform  it  when  the  days  are  12  hours  long  ? 

7.  A  man  sold  8  yards  of  cloth  for  7$  dollars  a 
yard,  and  received  8  firkins  of  butter  at  6f  dollars  a 
firkin  ;  how  much  was  then  due  to  him  ? 

8.  Two  men  are  38  miles  apart,  and  are  travelling 
awards  each  other,  one  at  the  rate  of  3  miles  an  hour, 
the  other  2  miles ;  how  much  do  they  approach  each 
other  in  an  hour  ?    How  much  in  2  hours  ?    In  how 
many  hours  will  they  meet  ?  At  what  distance  from 
each  place  from  which  they  set  out  ? 


Sect.  1 3.]  ARITHMETIC.  1  ]  5 

SECTION  XIII. 

A.  1.  If  you  give  J  of  an  orange  to  one  boy,  and 
J  to  another,  how  much  more  do  you  give  the  first 
than  the  second  ? 

2.  J  of  an  orange  is  how  many  J  of  an  orange  ? 

3.  If  you  give  J  of  an  orange  to  one  boy,  and  £  to 
another,  how  many  ^  would  you  give  away  ?    How 
many  J  would  you  have  left  ? 

4.  J  and  J  are  how  many  £  ? 

5.  A  man  gave  to  one  laborer  J  of  a  bushel  of 
wheat,  and  f  to  another ;  how  many  £  of  a  bushel  did 
he  give  to  both  ?  How  many  bushels  ? 

6.  J  and  f  are  how  many  £  ?  How  many  times  1  ? 

7.  A  man  gave  ^  of  a  barrel  of  flour  to  one  man, 
and  J  of  a  barrel  to  another ;  to  which  did  he  give 
the  most  ? 

8.  J  is  how  many  J  ? 

9.  A  man  bought  ^  of  a  bushel  of  wheat  at  one 
time,  and  f  of  a  bushel  at  another  ;  at  which  time  did 
he  buy  the  most? 

10.  ^  is  how  many  J  ? 

11.  A  man  bought  §  of  a  yard  of  cloth  at  one  time, 
and  f  of  a  yard  at  another  ;  at  which  time  did  he  buy 
the  most  ? 

12.  §  are  how  many  J  ? 

13.  A  man  wished  to  give  J  of  a  bushel  of  wheat 
to  one  man,  and  J  of  a  bushel  to  another ;  but  he 
could  not  tell  how  to  divide  it.    Another  man  standing 
by  advised  him  to  divide  the  whole  bushel  into  six 
equal  parts  first,  and  then  take  J  of  them  for  one,  and 
^  of  them  for  the  other.     How  many  parts  did   he 
give  to  each  ?  How  many  to  both  ?  How  many  had  he 
left  ? 

14.  J  is  how  many  J  ?  J  is  how  many  J  ?  \  and  \ 
are  how  many  \  ? 

15.  A  man  paying  some  money  to  his  laborers, 


1 1 6  ARITHMETIC.  [Part  1 

gave  each  man  J  of  a  dollar,  and  each  boy  £  of  a  dol- 
lar ;  how  much  more  did  he  give  to  a  man  than  to  a 
boy? 

16.  What  is  the  difference  between  £  and  J  ? 

17.  If  a  man  can  earn  J  of  a  dollar  in  a  day,  and  a 
boy  £  of  a  dollar,  how  much  does  the  man  earn  more 
than  the  boy  ? 

18.  What  is  the  difference  between  j  and  £  ? 

19.  A  boy  distributing  some  nuts  among  his  com- 
panions, gave  J  of  a  quart  to  one,  and  J  of  a  quart  to 
another ;  how  much  more  did  he  give  to  one,  than  to 
the  other  ? 

Note.      Change  them  to  sixths. 

20.  What  is  the  difference  between  £  and  J  ? 

21.  A  man  having  two  bushels  of  grain  to  distrib- 
ute among  his  laborers,  wished  to  give  J  of  a  bushel 
to  one,  and  f  of  a  bushel  to  another,  and  the  rest  to 
a  third  ;  but  was  at  a  loss  to  tell  how  to  divide  it ;  at 
last  he  concluded  to  divide  each  bushel  into  six  equal 
parts,  or  sixths,  and  then  to  distribute  those  parts. 
How  many  sixths  did  he  give  to  each  ? 

22.  §  is  how  many  J  ? 

23.  A  man  had  a  horse,  and  a  cow,  and  a  sheep. 
The  horse  would  eat  §  of  a  load  of  hay  in  the  winter, 
the  cow  J,  and  the  sheep  J.     How  many  J  of  a  load 
would  each  eat  ?     How  many  J  would  they  all  eat  ? 
How  many  loads  ? 

24.  A  boy  having  a  quart  of  nuts,  wished  to  divide 
them,  so  as  to  give  one  companion  J,  another  J,  and  a 
third  \  of  them  ;  but  in  order  to  make  a  proper  divis- 
ion, he  first  divided  the  whole  into  eif^t  equal  parts, 
and  then  he  was  able  to  divide  them  as  he  wished. 
How  many  eighths  did  he  give  to  each  ?    How  many 
eighths  had  he  left  for  himself? 

25.  J  is  how  many  \  ?  %  is  how  many  £  ?  £  and  J 
and  |  are  how  many  £  ? 

26.  A  man  gave  J  of  a  barrel  of  flour  to  one  man, 


Sect.  13.]  ARITHMETIC.  117 

and  f  of  a  barrel  to  another ;  to  which  did  he  give  the 
most  ?     How  much  ? 

27.  Which  is  the  larger,  f  or  £  ?     How  much  the 
larger  ? 

28.  A  boy,  having  a  pound  of  almonds,  said  'he  in- 
tended giving  J  of  them  to  his  sister,  and  J  to  his  broth- 
er, and  the  rest  to  his  mamma.     His  mamma  smiling 
said  she  did  not  think  he  could  divide  them  so.     Oh, 
yes,  I  can,  said  he,  I  will  first  divide  them  into  twelve 
equal  parts,  and  then  I  can  divide  them  well  enough. 
Pray  how  many  twelfths  did  he  give  to  each  ? 

29.  J  is  how  many  ^  ?  £  is  how  many  ^  ?  J  and 
J  are  how  many  -^  ? 

30.  Mr.  Goodman  having  a  pound  of  raisins,  said 
he  would  give  Sarah  J,  and  Mary  ^,  and  James  £  of 
them,  and  he  told  Charles  he  should  have  the  rest,  if 
he  could  tell  how  to  divide  them.     Well,  said  Charles, 
I  would  first  divide  the  whole  into  twelve  equal  parts, 
and  then  I  could  take  ^  and  £  and  J  of  them.     How 
many  twelfths  would  each  have  ? 

31.  J  and  £  and  J  are  how  many  ^  ? 

32.  George  bought  a  pine-apple,  and  said  he  would 
give  |  of  it  to  his  papa,  and  f  to  his  mamma,"  and  y3^ 
to  his  brother  James,  if  he  could  divide  it.  James  took 
it,  and  cut  it  into  twenty  equal  pieces,  and  then  distrib- 
uted them  as  George  had  desired.      How  many  twen- 
tieths did  he  give  to  each  ? 

33.  £  is  how  many  ^  ?  £  is  how  many  ^  ?  §  is 
how  many  ^  ?  T^  is  how  many  ^  ? 

34.  J  is  how  many  Tij  ? 

35.  J  is  how  many  ^  ? 

36.  ^  is  how  many  \  ? 

37.  £  is.  how  many  ^  ?    • 

38.  §  are  how  many 

39.  §  are  how  many 

40.  £  is  how  many  - 

41.  f  are  how  many 


118  ARITHMETIC.  [Parti. 

42.  £  are  how  many  T^  ? 

43.  f  are  how  m&ny  ^  ? 

44.  £  are  how  many  ^  ? 

45.  |  are  how  many  -fa  ? 

46.  f  are  how  many  ^  ? 

47.  ^y  are  how  many  ^  ? 

48.  Reduce  £  to  sixths  and  £  to  sixths. 

49.  §  and  §  are  how  many  £  ? 

50.  Reduce  £  and  £  to  eighths. 

51.  £  and  £  are  how  many  |  ? 

52.  J  and  £  are  how  many  £  ? 

53.  §  and  £  are  how  many  £  ? 

54.  £  and  f  are  how  many  J  ? 

55.  £  and  f  are  how  many  £  ? 

56.  £  and  §  are  how  many  T\j  ? 

57.  £  and  £  and  |  are  how  many  |  ? 

58.  J  and  f  and  T^  are  how  many  ^  ? 

59.  §  and  £  are  how  many  ^  ? 

60.  §  and  I  and  £  are  how  many  ^  ? 

61.  §  and  ^  and  £  are  how  many  ^  ? 

62.  £  and  £  and  f  and  J  and  T^  are  how  many^  ? 

63.  ^  and  ^  are  how  many  T^  ? 

64.  §  and  ^  are  how  many  T^  ? 

65.  §  and  §  are  how  many  -^  ? 

66.  §  less  £  are  how  many  |  ? 

67.  f  and  f ,  less  ^,  are  how  many  T\j  ? 

68.  £  less  f  are  how  many  ^  ? 

69.  £  less  f  are  how  many  ^  ? 

70  £  less  f  are  how  many  ^  ? 

71  £,  and  f ,  and  £,  and  ^,  less  f ,  are  how  many  ^  ? 

72.  i,  and  £,  and  f ,  and  ^,  and  ^y,  less  |,  are  how 
many  ^  ? 

73.  |  and  f  are  how  many  ^  ? 

74.  §  and  f  are  how  many  3^  ? 

75.  f  and  f  are  how  many  ^  ? 

When  the  denominators  in  two  or  more  fractions  are 


.Sect.  13.]  ARITHMETIC.  119 

the  same,  the  fractions  are  said  to  have  a  common 
denominator.  Thus  f  and  f  have"  a  common  denomina- 
tor. We  have  seen  that,  when  two  or  more  fractions 
have  a  common  denominator,  they  may  be  added  and 
subtracted  as  well  as  whole  numbers.  We  add  or  sub- 
tract the  numerators,  and  write  their  sum  or  difference 
over  the  common  denominator.  The  first  part  of  the 
process  in  the  above  examples  was  to  reduce  them  to 
a  common  denominator. 

76.  Reduce  f  and  f  to  a  common  denominator. 
Note.     They  may  be  reduced  to  twelfths. 

If  it  cannot  be  immediately  seen  what  number  must 
be  the  common  denominator,  it  may  be  found  by  mul- 
tiplying all  the  denominators  together ;  for  that  will 
always  produce  a  number  divisible  by  all  the  denomi- 
nators. 

77.  Reduce  f  and  f  to  a  common  denominator. 

78.  Reduce  §  and  f  and  }  to  a  common  denominator. 

79.  Reduce-  £  and  f  to  a  common  denominator. 

80.  Reduce  f  and  f  to  a  common  denominator. 

81.  Reduce  |-  and  f  and  f  to  a  common  denominator. 

82.  Add  together  f  and  f . 

83.  Add  together  f  and  T%. 

84.  Add  together  f  and  T%. 

85.  Add  together  J  and  f  and  fa 

86.  Subtract  ^  from  J. 

87.  Subtract  ^  from  f . 

88.  Subtract  f  from  f . 

89.  Subtract  f  from  £. 

B.  1.  Mr.  F.  said  he  would  give  £  of  a  pine-apple 
to  Fanny,  and  f  to  George,  and  the  rest  to  the  one  that 
could  tell  how  to  divide  it,  and  how  much  there  would 
be  left.  But  neither  of  them  could  tell ;  so  he  kept  it 
viimself.  Could  you  have  told,  if  you  had  been  there  ? 
How  would  von  divide  it  ?  How  much  would  be  left  ? 


120  ARITHMETIC.  [Part  \ 

2.  A  man  sold  1£  bushels  of  wheat  to  one  man,  4vj 
bushels  to  another ;  how  many  bushels  did  he  sell  to 
both? 

3.  A  man  bought  6£  bushels  of  wheat  at  one  time, 
and  2£   at   another ;  how  much   did  he  buy  in  the 
whole  ? 

4.  A  man  bought  7|  yards  of  one  kind  of  cloth,  and 
6J  yards  of  another  kind  ;    how  many  yards  in  the 
whole  ?    • 

5.  A  man  bought  f  of  a  barrel  of  flour  at  one  time, 
2£  barrels  at  another,  and  6|  at  another  ;   how  much 
did  he  buy  in  the  whole  ? 

6.  A  man  bought  one  sheep  for  4$  dollars,  and  an- 
other for  5£  dollars  ;  how  much  did  he  give  for  both  ? 

7.  There  is  a  pole  standing,  so  that  J  of  it  is  in  the 
mud,  and  §  of  it  in  the  water,  and  the  rest  out  of  the 
water ;  how  much  of  it  is  out  of  the  water  ? 

8.  A  man  having  undertaken  to  do  a  piece  of  work, 
did  i  of  it  the  first  day,  J  of  it  the  second  day,  and  £ 
of  it  the  third  day,  how  much  of  it  did  he  do  in  three 
days  ? 

9.  A  man  having  a  piece  of  work  to  do,  hired  two 
men  and  a  boy  to  do  it.     The  first  man  could  do  J  of 
the  work  in  a  day,  and  the  other  £  of  it,  and  the  boy 
£  of  it ;  how  much  of  it  would  they  all  do  in  a  day  ? 

Note.  By  dividing  a  line  into  halves,  %  and  then  into 
fourths,  it  will  be  seen  that  £  is  the  same  as  £  ;  a  line  di- 
vided into  halves  and  then  into  sixths,  will  show  that  £  is 
the  same  as  £,  and  4  as  f  ;  f ,  $,  can  therefore  be  reduced 
to  -J,  and  |  to  f .  This  is  called  reducing  fractions  to  their 
lowest  terms.  It  is  done  by  dividing  by  the  greatest  num- 
ber that  will  divide  it  without  a  remainder. 

C.  1.  Reduce  f  to  its  lowest  terms.*     Am.  f . 

2,  Reduce  ^  to  its  lowest  terms. 

3.  Reduce  f  to  its  lowest  terms. 

*  If  this  article  should  be  found  too  difficult  for  the  pupil,  he  may 
Dmit  it  till  after  the  next  section. 


Sect.  14.]  ARITHMETIC.  121 

4.  Reduce  fa  to  its  lowest  terms. 

5.  Eeduce  |£  to  its  lowest  terms. 

6.  Reduce  ^  to  its  lowest  terms. 

7.  Reduce  ^  to  its  lowest  terms. 

8.  Reduce  ^  to  its  lowest  terms. 

9.  Reduce  Jf  to  its  lowest  terms. 

10.  Reduce  •£$  to  its  lowest  terms. 

11.  Reduce  §f  to  its  lowest  terms. 

12.  Reduce  §J  to  its  lowest  terms. 

13.  Reduce  f  §  to  its  lowest  terms. 

14.  Reduce  f  \  to  its  lowest  terms. 

Note.  It  will  be  seen  by  the  above  section  that  ii 
both  the  numerator  and  denominator  be  multiplied  by 
the  same  number,  the  value  of  the  fraction  will  not  be 
altered ;  or  if  they  can  both  be  divided  by  the  same 
number  without  a  remainder,  the  fraction  will  not  be 
altered. 


SECTION   XIV. 

A.  1.  A  BOY  having  J  of  an  orange  gave  away  ^ 
of  that,  what  part  of  the  whole  orange  did  he  give 
away  ? 

2.  What  is  i  of  J  ? 

3.  If  you  cut  an  apple  into  three  pieces,  and  then 
cut  each  of  those  pieces  into  two  pieces,  how  many 
pieces  will  the  whole  apple  be  cut  into  ?     What  part 
of  the  whole  apple  will  one  of  the  pieces  be  ? 

4.  What  is  J  of  J  ? 

5.  A  boy  had  J  of  a  pine  apple,  and  cut  that  half 
into  three  pieces,  in  order  to  give  away  ^  of  it.  What 
part  of  the  whole  apple  did  he  give  away  ? 

6.  What  is  £  of  £  ? 

7.  If  an*  orange  be  cut  into  4  parts,  and  then  each 
of  the  parts  be  cut  in  two,  how  many  pieces  will  the 
whole  be  cut  into  ? 

8.  What  is  4  of  J  ? 


1 22  ARITHMETIC.  [Part  1. 

9.  A  man  having  J  of  a  barrel  of  flour,  sold  |  of 
that ;  how  much  did  he  sejil  ? 

10.  What  is  i  of  J? 

11.  If  an  orange  be  cut  into  4  equal  parts,  and 
each  of  those  parts  be  cut  into  3  equal  parts,  how 
many  parts  will  the  whole  orange  be  cut  into  ? 

12.  What  is  J  of  J  ? 

13.  A  boy  having  £  of  a  quart  of  chestnuts,  gave 
away  J  of  what  he  had.     What  part  of  the  whole 
quart  did  he  give  away  ? 

14.  What  is  £  of  £  ? 

15.  What  is  J  of  J  ? 

1 6.  A  man  owning  J  of  a  ship,  sold  ^  of  his  share : 
what  part  of  the  ship  did  he  sell,  and  what  part  did 
he  then  own  ? 

17.  What  is  J  of  J  ? 

18.  What  is  J  of  i  ? 

19.  What  is  J  of  J  ? 

20.  What  is  J  of  £  ? 

21.  What  is  |  of  |  ? 

22.  What  is  £  of  £  ? 

23.  What  is  |  of  |  ? 

24.  What  is  J  of  £  ? 

25.  What  is  |  of  |  ? 

26.  What  is  £  of  j»f 

27.  What  is  £  of  £  ? 

28.  What  is  }  of  \  ? 

29.  A  boy  having  §  of  an  orange,  (that  is,  2  pieces,) 
gave  his  sister  J  of  what  he  had ;  how  many  thirds 
did  he  give  her  ? 

30.  What  is  J  of  §  ? 

31.  A  boy  having  f  of  a  pine  apple,  said  he  would 
give  one  half  of  what  he  had  to  his  sister,  if  she 
could  tell  how  to  divide  it.     His  sister  says,  you  have 
got  f ,  or  three  pieces,  if  you  cut  them  all  in  two,  you 
can  give  me  J  of  them.     But  J  of  J  is  J,  therefore  i 
shall  have  f  of  the  whole  pine  apple. 


Sect.  14.]  ARITHMETIC.  123 

32.  What  is  J  of  |  ? 

33.  A  man  owning  f   of  a  share  in  the  Boston 
Bank,  sold  J  of  his  part.     What  part  of  a  share  did 
he  sell  ? 

34.  What  is  J-  of  f  ? 

35.  A  man  owning  f  of  a  ship,  sold  ^  of  his  share ; 
what  part  of  the  whole  ship  did  he  sell  ?     What  part 
had  he  left  ? 

36.  What  is  J  of  J  ? 

37.  What  is  J  of  |  ? 

38.  What  is  |  of  f  ? 
39, -What  is  |  of  f  ? 

40.  What  is  J  of  V2-  ? 

41.  What  is  J  of  ?  ? 

42.  A  man  owning  f  of  a  share  in  the  Boston 
Bank,  sold  J  of  his  part ;  what  part  of  a  whole  share 
did  he  sell  ? 

43.  What  is  J  of  f  ? 

44.  What  is  J  of  f  ? 

45.  A  boy  having  £  of  a  water-melon,  wished  to 
divide  his  part  equally  between  his  sister,  his  brother, 
and  himself,  but  was  at  a  loss  to  know  how  to  do  it ; 
but  his  sister  advised  him  to  cut  each  of  the  fifths  into 
3  equal  parts.     How  many  pieces  did  each  have  ?  and 
what  part  of  the  whole  melon  was  each  piece  ? 

46.  What  is  J  of  £  ? 

47.  What  is  J  of  |  ? 

48.  What  is  J  of  f  ? 

49.  What  is  £  of  f  ? 

50.  What  is  ^y  of  |  ? 

51.  What  is  J  of  J  ? 

52.  What  is  §  of  i  ? 

53.  What  is  J  of  §  ? 

54.  What  is  |  of  |  ? 

55.  What  is  £  of  f  ? 

56.  What  is  |  of  f  ? 

57.  What  is  J  of  f  ? 


124  ARITHMETIC.  [Part  1 

58.  What  is  §  of  g  ? 

59.  What  is  £  of  |  ? 

60.  What  is  \  of  f  ? 

61.  What  is  f  of  §  ? 

62.  What  is  |  of  f  ? 

63.  What  is  f  of  f  ? 

64.  What  is  ^  of  }  ? 

65.  What  is  ^  of  }  ? 

66.  What  is  |  of  ^  ? 

67.  What  is  f  of  §  ? 

68.  What  is  £  of  f-  ? 

69.  If  a  yard  of  cloth  cost  2J  dollars,  what  will  \ 
of  a  yard  cost  ? 

70.  What  is  J  of  2  J  ? 

71.  A  boy  had  2£  oranges,  and  wished  to  givjB  J  of 
them  to  his  sister,  and  £  to  his  brother,  but  he  did  not 
know  how  to  divide  them  equally.     His  brother  told 
him  to  cut  the  whole  into  halves,  and  then  cut  each 
of  the  halves  into  3  pieces.     What  part  of  a  whole 
orange  did  each  have  ? 

72.  What  is  J  of  2  J  ? 

73.  A  man  bought  4  bushels  of  corn  for  3§  dollars ; 
what  part  of  a  dollar  did  1  bushel  cost  ? 

Change  the  3|  to  thirds,  and  then  find  J  of  y  as 
above. 

74.  What  is  i  of  5§  ? 

75.  If  5  bushels  of  wheat  cost  7J  dollars,  what  is 
that  a  bushel  ? 

7£.  What  is  J-  of  7f  ? 

77.  A  man  bought  6  gallons  of  alcohol  for  8f  dol- 
lars ;  what  was  that  a  gallon  ? 

78.  What  is  J  of  8§  ? 

79.  A  man  bought  7  gallons  of  wine  for  8f  dollars 
how  much  was  that  a  gallon  ? 

80.  What  is  }  of  8f  ? 

81.  A  man  bought  10  pieces  of  nankin  for  6|  dol- 
lars ;  how  much  was  it  a  piece  ? 


Sect.  H.]  ARITHMETIC.  125 

82.  What  is  ^  of  6|  ? 

83.  If  9  bushels  of  rye  cost  7f  dollars,  what  is  that 
a  bushel  ? 

84  What  is  £  ef  7%  ? 

85.  What  is  J  of  5f  ? 

86.  What  is  |  of  8j  ? 

87.  What  is  i  of  6  A  ? 

88.  What  is  £  of  9£  ? 

89.  A  man  bought  7  yards  of  cloth  for  18  J  dollars  ; 
what  was  that  a  yard  ?     What  would  3  yards  cost  at 
that  rate  ? 

90.  What  is  |  of  18  J  ?     What  is  f  of  18§  ? 

91.  A  man  bought  5  barrels  of  cider  for  27§  dol- 
lars ;  what  was  it  a  barrel  ?     What  would  7  barrels 
cost  at  that  rate  ? 

92.  What  is  j-  of  27|  ?    What  is  J  of  27f  ? 

93.  If  6  barrels  of  flour  cost  38f  dollars,  what  would 
10  barrels  cost  at  that  rate  ? 

94.  What  is  V°  of  38$  ? 

B.  1.  A  man  bought  a  piece  of  cloth  for  42f  dol- 
lars, and  was  obliged  to  sell  it  for  £  of  what  it  cost 
him  ;  how  much  did  he  lose  ? 

2.  A  man  bought  a  quantity  of  flour  for  53f  dol- 
lars, and  sold  it  for  f  of  what  it  cost  him ;  how  much 
did  he  gain  ? 

.  3.  If  7  men  can  do  a  piece  of  work  in  4f  days,  how 
long  will  it  take  1  man  to  do  it  ?  How  long  will  it 
take  3  men  to  do  it? 

4.  If  4  men  can  do  a  piece  of  work  in  9f  days, 
how  long  would  it  take  to  do  it 'if  7  men  were  em 
ployed  ? 

5.  There  is  a  pole  standing  so  that  f  of  it  is  in  the 
water,  and  §  as  much  in  the  mud ;  how  much  is  in 
the  mud  ? 

6.  If  a  man  can  travel  13f  miles  in  3  hours,  how 
many  miles  will  he  travel  in  8  hours  ? 


126  ARITHMETIC.  [Part  1. 

7.  If  5  horses  will  eat  26f  loads  of  hay  in  a  year, 
what  will  8  horses  eat  in  the  same  time  ? 

8.  If.  4  pipes  will   empty  a   cistern   in    6g  hours, 
how  long  will   it  take  7  pipes  of.  the  same  size  tc 
empty  it?. 

SECTION  XV. 

A.  1.  A  BOY  having  2  oranges  wished  to  give  £  of 
an  orange  apiece  to  his  playmates  ;  how  many  could 
he  give  them  to  ?  If  he  had  given  §  of  an  orange 
apiece,  how  many  could  he  have  given  them  to  ? 

2.  How  many  times  £  are  there  in  2  ?     How  many 
times  §  are  there  in  2  ? 

3.  A  man  having  3  bushels  of  corn,  distributed  it 
among  some  poor  persons,  giving  them  f  of  a  bushel 
each  ;  to  how  many  did  he  give  it  ? 

Note.  First  jind  how  many  he  would  have  given 
it  to,  if  he  had  given  £  of  a  bushel  to  each. 

4.  In  3  are  how  many  times  ^  ?  How  many  times  f  ? 

5.  If  f  of  a  barrel  of  flour  will  last  a  family  one 
month,  how  long  will  4  barrels  last  the  same  family  ? 
How  long  will  6  barrels  last  ?      How  long  will  10 
barrels  last  ? 

6.  How  many  times  is  £  contained  in  4  ?      How 
many  times  in  6  ?     How  many  times  in  10  ? 

7.  If  f  of  a  bushel  of  wheat  will  last  a  family  one 
week,  how  many  weeks  will  6  j  bushels  last  the  same 
family  ? 

8.  How  many  times  is  f  contained  in  6  J  ? 

9.  There  is  a  cistern  having  a  pipe  which  will  fill  it 
in  f  of  an  hour ;  how  many  tunes  would  the  pipe  fill 
the  cistern  in  3|  hours  ? 

10.  How  many  times  is  |  contained  in  3f  ? 

11.  How  much  cloth  at  1^  dollars  (that  is  f  dollars) 
a  yard  can  be  bought  for  4  dollars  ? 

12.  How  many  times  is  1 J  or  |  contained  in  4  ? 


Sect.  15.]  ARITHMETIC.  127 

13.  A  man  distributed  8^  bushels  of  wheat  among 
Borne  poor  persons,  giving  1  £  bushels  to  each ;  how 
many  did  he  give  it  to  ? 

14.  How  many  times  is  1^  contained  in  8J  ? 

15.  If  a  soldier  is  allowed  1£  pounds  (that  is  j  of  a 
pound)  of  meat  in  a  day,  to  how  many  soldiers  would 
6 1  pounds  be  allowed  ? 

16.  How  many  times  is  1^  contained  in  6§? 

17.  If  If  tons  of  hay  will  keep  a  horse  through 
the  winter,  how  many  horses  will  10  tons  keep  ? 

18.  How  many  times  is  If  contained  in  10  ? 

19.  At  2^  dollars  a  box,  how  many  boxes  of  raisins 
can  be  bought  for  10  dollars  ? 

20.  How  many  times  is  2^  contained  in  10  ? 

21.  At  If  dollars  a  pound,  how  many  pounds  of 
indigo  can  be  bought  for  9f  dollars  ? 

22.  How  many  times  is  If  contained  in  9f  ? 

23.  At  If  dollars  a  barrel,  how  many  barrels  of 
raisins  can  be  bought  for  9f  dollars  ? 

24.  How  many  times  is  If  contained  in  9f  ? 

25.  At  §  of  a  dollar  apiece,  how  many  pieces  of 
nankin  can  be  bought  for  8|  dollars  ? 

26.  How  many  times  is  £  contained  in  8f  ? 

27.  At  §  of  a  dollar  a  pound,  how  many  pounds  of 
tea  can  be  bought"  for  7f  dollars  ? 

28.  How  many  times  is  f  contained  in  7§  ? 

29.  How  many  times  is  3^  contained  in  7|  ? 

30.  How  many  times  is  5  £  contained  in  17  ?     . 

31.  How  many  times  is  4£  contained  in  9f  ? 

32.  Plow  many  times  is  3f  contained  in  12f  ? 

B.  1.  At  y1^  of  a  dollar  a  pound,  how  many  pounds 
of  meat  can  be  bought  for  £  of  a  dollar  ? 

Note.      Change  ^  to  tenths. 

2.  How  many  times  is  yV  contained  in  J  ? 

8.  A  man  having  f  of  a  barrel  of  flour,  distributed 
it  among  some  poor  persons,  giving  them  £  of  a  barrel 
apiece  ;  hew  many  did  he  give  it  to  ? 


128  ARITHMETIC.  [Par/  1. 

Note.      CJiange  both  fractions  to  twelfths ;  that  t'j, 
reduce  them  to  a  common  denominator. 

4.  How  many  times  is  %  contained  in  J  ? 

5.  If  a  pound  of  almonds  cost  f  of  a  dollar,  hovr 
many  pounds  can  be  bought  for  §  of  a  dollar  ? 

Note.    Reduce  the  fractions  to  a  common  denomi- 
nator. 

6.  How  many  times  is  }  contained  in  §  ? 

7.  If  a  piece  of  nankin  cost  §  of  a  dollar,  how  many 
pieces  can  be  bought  for  4£  dollars  ?  that  is,  ^  dol- 
lars ? 

8.  How  many  times  is  §  contained  in  4f  ? 

9.  If  a  bushel  of  barley  cost  £  of  a  dollar,  how 
many  bushels  can  be  bought  for  £  of  a  dollar  ?     How 
many  for  1  f  dollars  ? 

10.  How  many  times  is  J  contained  in  f  ?     How 
many  times  in  If  ? 

11.  How  many  times  is  $  contained  in  §  ? 

12.  How  many  times  is  §  contained  in  f  ? 


TABLES  OF  COINS,  WEIGHTS,  AND  MEASURES. 

Ttible  I.  —  FEDERAL  MONEY. 
Federal  Money  is  the  currency  of  the  IJnited  States. 

10  mills  (m.)  make  1  cent,  marked  ct. 

10  cents  "      1  dime                     "       d. 

10  dimes  "      1  dollar,                   "       doll  or  $ 

10  dollars  "      1  Eagle,                  "       E. 

Table  II.  —  STERLING  MONEY. 
Sterling  Money  is  the  currency  of  England. 

4  farthing?  (qr.  or  far.)  make  1  penny,                       marked  d. 

12  pence  "      1  shilling,                                   «. 

20  shillings  "      1  pound,  or  a  sovereign,   "       £ 

W  shillings  "      1  guinea. 


meet.  15.]  ARITHMETIC.  129 

Table  III.  —  TROY  WEIGHT. 

Used  in  weighing  Gold,  Silver,  and  some  Liquids. 

24  grains  (gr.)  make  1  pennyweight,         marked  dwi 

20  pennyweights  "       1  ounce,  "        oz. 

12  ounces  "      1  pound,  "         Ib 

Table  IV.  —  AVOIRDUPOIS  WEIGHT. 

Used  in  weighing  Groceries,  Hay,  &c.,  and  all  the 
Metals,  except  Gold  and  Silver.  It  is  now  the  custom  to 
allow  100  Ibs.  for  a  hundred-weight  instead  of  112  as  for- 
merly. 


16  drams  (dr.) 
16  ounces 
25  pounds 

4  quarters 
20  hundred-weight 


make  1  ounce,  marked  oz. 

1  pound,  "      U>. 

1  quarter,  "  qr. 

"  1  hundred-weight,  "  cwt. 

"  Iton,  "  T. 


Table  V.  —  CLOTH  MEASURE. 

Used  in  measuring  Woollens  and  other  Cloths, 
inches  (in.)  make  1  nail, 


4  nails,  or  9  in. 

4  quarters 
3  quarters 

5  quarters 

6  quarters 


marked  na. 

1  quarter  of  a  yard,    "        qr. 
1  ya*d,  "        yd. 

1  Flemish  ell,  "        Fl  e. 

1  English  ell,  "         E.  e. 

1  French  ell,  "         F.  e. 


Table  VI.  —  WENE  MEASURB. 
Used  in  measuring  Wines,  &c. 


4  gills  (gl) 

2  pints 

4  quarts 
31 2  gallons 
42  gallons 
63  gallons,  or  2  barrels 

2  hogsheads 

2  pipes 


Table  Vll.  —  DRY  MEASURE. 

Used  in  measuring  Grains,  Salt,  Oysters,  &c. 

2  pints  make  1  quart,  marked  qt. 

8  quarts  "      1  peck,  "       pk. 

4  pecks  "      1  bushel,  «       fo*. 


ake  1  pint, 
"      1  quart, 
"      1  gallon, 
1  barrel, 
1  tierce, 
1  hogshead, 
1  pipe  or  butt, 
1  tun, 

marked  pi. 

«          qt. 
"         gal 
bar.  or  bbl. 
"        tier. 
"        hhd. 
"        pi. 
"        tun. 

180  AEITHMETIC.  [Part  I 

Table  VIII.  —  MEASURE  OF  TIME. 


minute,  marked  min. 

hour,  "        hr. 

day,  "        dy. 

k.  "         wk. 


wee 
nion 


. 
tn, 


60  seconds  (sec.)  make 

60  minutee 

24  hours  " 

7  days 

4  weeks  " 

13  months,  1  day  and  6  J 
hours,  or  12  calendar  >       "      1  year,  "        yr. 

months 

For  convenience  of  reckoning,  it  is  usual  in  calen- 
dars to  call  the  year  365  days  for  3  successive  years, 
and  every  fourth  year  366,  (for  in  4  years  the  six 
hours  overplus  amount  to  a  day,)  which  is  called 
bissextile  or  leap  year.  This  day  is  added  to  Feb- 
ruary. 

The  common  year  is  divided  into  twelve  months, 
which  are  sometimes  called  calendar  months,  because 
they  are  the  months  used  in  calendars. 

The  names  of  the  months,  and  the  number  of  days 
in  each  are  as  follows : 

Names.  Number  of  days. 

1.  January  31 

2.  February  28,  in  leap  year  29 

3.  March  31 

4.  April  30 

5.  May  31 

6.  June  30 

7.  July  31 

8.  August  31 

9.  September  30 

10.  October  31 

11.  November  30 

12.  December  31 

Miscellaneous  Example*. 

1.  In  2  pounds  how  many  ounces  ? 

2.  In  8  yards  how  many  quarters  ? 


Sect.  15.]  ARITHMETIC.  131 

3.  ID  3  quarters  of  a  yard  how  many  nails  ? 

4.  $j  of  a  dollar  is  how  many  cents  ? 

5.  How  many  farthings  is  g  of  a  penny  ? 

6.  How  many  pence  is  $  of  a  shilling  ? 

7.  |  of  a  yard  is  how  many  quarters  and  nails  ? 

8.  In  £f  how  many  shillings  ? 

9.  How  much  is  f  of  a  shilling  ? 

1 0.  How  much  is  f  of  a  bushel  of  wheat  ? 

11.  How  much  would  f  of  a  barrel  of  wine  cost,  at 
one  dollar  a  gallon  ? 

12.  How  much  would  J  cwt.  of  sugar  cost,  at  8 
cents  a  pound  ? 

13.  How  much  is  f  of  a  day  ?       » 

14.  How  much  is  |  of  a  day  ? 

15.  How  much  is  f  of  a  week  ? 

16.  How  much  is  f  of  an  hour  ? 

17.  How  much  would  f  of  a  hogshead  of  wine  cost, 
at  2  dollars  a  gallon  ? 

18.  If  a  man  spend  28  dollars  in  a  month,  what  is 
that  a  week  ?    How  much  a  day  ? 

19.  If  a  man  spend  16  dollars  a  week,  what  is  that 
a  day? 

20.  If  a  man  buy  4  bushels  of  grain  for  5  dollars, 
how  much  is  that  a  bushel  ? 

21.  If  wine  is  2  dollars  a  gallon,  how  much  is  that 
a  pint  ? 

22.  If  you  give  5  cents  a  gill  for  wine,  what  is 
that  a  pint  ?  What  is  it  a  quart  ?  What  is  it  a  gal- 
lon? 

23.  If  wine  is  worth  20  cents  a  pint,  what  is  that  a 
gill  ?  What  is  it  a  quart  ?  What  is  it  a  gallon  ? 

24.  If  a  yard  of  cloth  is  worth  7  dollars,  what  are 
2 1  yards  worth? 

25.  If  a  man  earn  11  dollars  a  week,  what  is  that 
a  day  ?  What  for  3  days  ?    What  for  4  J  days  ? 

26.  If  a  man  earn  2f  dollars  in  a  day,  what  will  he 
earn  in  a  week  ? 


132  ARITHMETIC.  [Part  1. 

27    What  is  £  of  a  hogshead  of  wine  ? 

28.  1  farthing  is  what  part  of  a  penny  ? 

29.  2  farthings  is  what  part  of  a  penny  ? 

30.  3  farthings  is  what  part  of  a  penny  ? 

31.  1  penny  is  what  part  of  a  shilling? 

32.  2  pence  is  what  part  of  a  shilling  ? 

33.  3  pence  is  what  part  of  a  shilling? 

34.  5  pence  is  what  part  of  a  shilling  ? 

35.  6  pence  is  what  part  of  a  shilling? 

36.  7  pence  is  what  part  of  a  shilling? 

37.  8  pence  is  what  part  of  a  shilling  ? 

38.  9  pence  is  what  part  of  a  shilling? 

39.  10  pence  is  what  part  of  a  shilling  ? 

40.  11  pence  is  what  part  of  a  shilling? 

41.  1  shilling  is  what  part  of  a  pound  ? 

42.  2  shillings  is  what  part  of  a  pound  ? 

43.  3  shillings  is  what  part  of  a  pound  ? 

44.  4  shillings  is  what  part  of  a  pound  ? 

45.  5  shillings  is  what  part  of  a  pound  ? 

46.  What  part  of  a  pound  is  6  shillings?     7  shil- 
lings ?     8  shillings  ?     9  shiUings  ?     10  shillings  ?     1 1 
shillings  ?    12  shillings  ?     13  shillings  ?    14  shillings  ? 

I  r>  shillings  ?  16  shillings  ?  17  shillings  ?  18  shillings  ? 
19  shiUings? 

47.  How  many  farthings  are  there  in  a  shilling? 

48.  One  farthing  is  what  part  of  a  shilling  ? 

49.  2  farthings  is  what  part  of  a  shilling  ?     3  far- 
things ?    4  farthings  ?    5  farthings  ?     6  farthings  ?     7 
farthings  ?    8  farthings  ?    9  farthings  ?     10  farthings  ? 

50.  How  many  pence  are  there  in  a  pound  ? 

51.  One  penny  is  what  part  of  a  pound  ? 

52.  What  part  of  a  pound  is  2  pence  ?     3  pence  ? 
4  pence  ?     5  pence  ?     6  pence  ?    7  pence  ?    8  pence : 

II  pence  ?    15  pence  ?    27  pence  ?    35  pence  ? 

53.  How  many  pence  are  there  in  I  shilling  and  6 
pence  ? 

54.  In  2  shillings  ard  4  pence,  how  many  pence  ? 


Sect.  15.]  ARITHMETIC.  133 

55.  In  4  shillings  and  5  pence,  how  many  ponce  ? 

56.  In  5  shillings  and  8  pence,  how  many  pence  ? 

57.  In  9  shillings  and  1 1  pence,  how  many  pence  i 

58.  What  part  of  £1  is  2s.  6d.  ? 

59.  3s.  5d.  is  what  part  of  £1  ? 
Note.     Reduce  the  whole  to  pence. 

60.  7s.  8d.  is  what  part  of  £1  ? 

61.  What  is  the  price  of  2  yards  of  cloth,  at  3s.  4d 
a  yard  ? 

62.  What  will  8  yards  of  cloth  cost,  at  2s.  8d.  a 
yard  ? 

63.  What  will  4  bushels  of  wheat  cost,  at  5s.  9d.  a 
bushel  ? 

64.  What  must  you  give  for  4  barrels  of  cider,  at 
2^  dollars  a  barrel  ? 

65.  If  3  bushels  of  wheat  be  divided  between  2 
men,  how  much  would  they  have  apiece  ? 

66.  If  4  bushels  of  corn  be  divided  among  5  men, 
how  much  would  they  have  apiece  ? 

67.  If  3  bushels  of  corn  be  divided  among  7  men, 
how  much  would  they  have  apiece  ? 

68.  How  many  nails  are  there  in  1  yard  ? 

69.  How  many  nails  are  there  in  4  yards  ? 

70.  How  many  nails  are  there  in  5  yards  and  2 
nails  ? 

71.  In  7  yards  and  3  quarters,  how  many  quarters  ? 

72.  In  4  yards,  2  quarters,  and  3  nails,  how  many 
nails  ? 

73.  1  nail  is  what  part  of  a  quarter  ? 

74.  3  nails  is  what  part  of  a  quarter  ? 

75.  1  nail  is  what  part  of  a  yard  ? 

7  6.  What  part  of  1  yard  is  3  nails  ?     5  nails  ?     7 
nails  ?     10  nails  ?     15  nails  ? 

77.  In  8  quarters  of  a  yard  how  many  yards  ? 

78.  In  12  quarters  of  a  yard  how  many  yards  ? 

79.  In  10  quarters  of  a  yard  how  many  yards  ? 

80.  In  15  quarters  of  a  yard  how  many  yards? 


134  ARITHMETIC.  [Part  1. 

81.  In  12  nails  how  many  quarters  of  a  yard  ? 

82.  In   10    nails    how  many  quarters  of  a  yard? 
How  many  yards  ? 

83.  In    24    uuils   how  many  quarters  of  a  yard? 
How  many  yards  ? 

84.  In  35  quarters  of  a  yard  how  many  yards  ? 

85.  In  45  nails  how  many  yards  ? 

86.  In  63  nails  how  many  yards  ? 

87.  At  2  cents  a  nail  what  would  4  yards  of  cloth 
cost? 

88.  At  2§  dollars  for  1  quarter  of  a  yard,  what 
would  2  yards  cost  ? 

89.  1  oz.  is  what  part  of  a  Ib.  ? 

90.  What  part  of  a  Ib.  is  2  oz.  ?    3  oz.  ?    4  oz.  ?    5 
oz.  ?  7  oz.  ?  10  oz.  ?  15  oz.  ? 

91.  What  part  of  a  qr.  of  1  cwt.  is  1  Ib.  ?    2  Ibs.  ? 
31bs.?    4  Ibs.?    7  Ibs.?    9  Ibs.  ?    14  Ibs.  ?    18  Ibs.? 
23  Ibs.  ? 

<r2.  At  3  cents  for  1  oz.  what  would  1  Ib.  cost  ? 

93.  At  3  cents  for  2  oz.  what  would  1  Ib.  cost  ? 

94.  At  3  cents  for  8  oz.  what  would  1  Ib.  cost  ? 

95.  At  5  cents  for  10  oz.  what  would  1  Ib.  cost? 

96.  At  8  shillings  for  4  Ibs.  what  would  10  Ibs 
cost? 

97.  If  a  man  consume  1  Ib.  and  3  oz.  of  meat  in  a 
day,  how  much  would  he  consume  in  a  week  ? 

98.  If  a  man  spend  2f  dollars  in  a  day,  how  much 
would  he  spend  in  a  week  ? 

99.  If  a  man  travel  3f  miles  in  an  hour,  how  far 
would  he  travel  in  3  hours  ?     How  far  in  7  hours  ? 
How  far  in  1 2  hours  ? 

100.  If    2    men    start  from  the  same    place,  and 
travel  in  opposite  directions  ;  one  at  the  rate  of  3| 
miles   in  an  hour,  and  the  other  4  J  miles ;  how  far 
will  they  be  apart  at  the  end  of  1  hour  ?     Plow  far  at 
the  end  of  2  hours  ?     How  far  at  the  end  of  3  hours  ? 
How  far  at  the  end  of  7  hours  ? 


Sect.  15.]  ARITHMETIC.  135 

101.  Two  men   start  from   the   same   place,   and 
travel  the  same  way ;  one  at  the  rate  of  4|  miles  in 
an  hour,  the  other  at  the  rate  of  4£  miles  in  an  hour ; 
how  far  will  they  be  apart  at  the  end  of  1   hour  ? 
How  far  in  2  hours  ?     How  far  in  5  hours  ?    How  fai 
in  10  hours  ?     How  far  in  3  days,  if  they  travel  10 
hours  in  a  day  ? 

102.  How  many  yards  of  cloth,  at  5  dollars  a  yard, 
must  fye  given  for  8  barrels  of  flour,  at  7  dollars  per 
barrel ? 

103.  What  part  of  a  month  is  1  week  ?  2  weeks  ? 
3  weeks? 

104.  What  part  of  a  year  is  1  month  ?  2  months? 

3  months  ?     4  months  ?     5  months  ?     6  months  ?     7 
months?    8  months?    9  months?    10  months?     11 
months  ? 

105.  What  part  of  1  month  is  1  day?    2  days?    3 
days  ?  7  days  ?  8  days  ?  11  days  ?  15  days  ?  18  days  ? 
20  days  ?  24  days  ?  27  days  ? 

106.  If  5  bushels  of  oats  will  keep  7  horses  through 
the  winter,  how  many  bushels  will  it  take  to  keep  12 
horses  the  same  time  ? 

107.  If  you  give  7  men  2£  bushels  of  corn  apiece, 
how  many  bushels  would  it  take  for  the  whole  ? 

108.  A  man,  failing  in  trade,  was  able  to  pay  his 
creditors    only   4  shillings    on   a    dollar ;    how  much 
would  he  pay  on  2  dollars  ?    How  much  on  3  dollars  ? 
How  much  on  7  dollars  ?     How  much  on  10  dollars  ? 

109.  A  man,  failing  in  trade,  is  able  to  pay  only  9 
shillings  on  a  pound  ;  how  much  would  he  pay  on  a 
debt  of  2  pounds  ?     How  much  on  3  pounds  ?     How 
much  on  12  pounds  ? 

110.  A  man,  failing  in  trade,  is  able  to  pay  only 

4  shillings  and  7  pence  on  a  dollar  ;  how  much  would 
he  pay  on  a  debt  of  7  dollars  ? 

111.  If  6  dollars'  worth  of  provision  will  serve  3 
men  5  days,  how  many  days  will  it  serve   1   man  \ 


136  ARITHMETIC.  [Part  L 

How  many  days  will  it  serve  2   men  ?    How  many 
days  will  it  serve  8  men  ? 

112.  If  10  dollars'  worth  of  provision  will  serve  7 
men  4  days,  how  many  days  will  it  serve  9  men  ? 

113.  If  12  dollars'  worth  of  provision  will  serve  5 
men  3  days,  how  many  men  would  it  serve    1   day  ? 
How  many  2  days  ?     How  many  8  days  ? 

114.  If  11  dollars'  worth  of  provision  will  servo  6 
men  8  days,  how  many  men  will  it  serve  5  days  '? 

115.  If  8  dollars'  worth  of  provision  will  serve  7 
men  5  days,  how  many  days  would  16  dollars'  worth 
of  provision  last  4  men  ? 

116.  If  1  peck  of  wheat  afford  12  ten-penny  loaves, 
how  many  penny  loaves  may  be    obtained    in  mi    it  ? 
How  many  two-penny   loaves  ?     How    many    three- 
penny loaves  ?  How  many  seven-penny  loaves  ? 

117.  If  1    peck    of  wheat    afford    11    eight-penny 
loaves,  how  many  ten-penny  loaves  will  it  afford  ? 

1 18.  A  man  having  hired  some  men  and  some  boys, 
agreed  to  give  each  man  3  shillings,  and  each  boy  2 
shillings  ;  how  much  would  it  fcike  to  pay  a  man  and 
a  boy  ?  How  much  2  men  and  2  boys  ?  How  much  7 
men  and  7  boys  ? 

,  119.  A  man  having  18  shillings  to  pay  among  his 
laborers,  would  give  to  every  man  2  shillings,  and  to 
every  boy  1  shilling ;  the  number  of  men  and  boys 
was  equal ;  how  many  were  there  of  each  ? 

120.  A  gentleman  having  50  shillings  to  pay  among 
his  laborers,  would  give  to  every  man  8  pence,  and  to 
every  boy  4  pence  ;  the  number  of  men  and  boys  was 
equal  ;  how  many  were  there  of  each  ? 

121.  Two  men  bought  a  bushel  of  corn  ;  one  gave  1 
shilling,  the  other  2  shillings ;  what  part  of  the  whole 
did  each  pay  ?  What  part  of  the  corn  must  each  have  ? 

122.  Two  men  bought  a  barrel  of  flour  for  8  dol- 
lars ;  one  gave  3  dollars,  the  other  5  dollars  ;  what 
oart,  did  each  pay  ?  and  what  part  must  each  have  ? 


Sect.  15.]  ARITHMETIC.  137 

123.  Three  men,  A,  B,  and  C,  hired  a  garden  :  A 
paid   6  dollars,  B   5   dollars,  and  C   9   dollars  ;  how 
much  did  they  all  pay  ?    What  part  of  the  whole  did 
each  pay  ?    They  sold    the    produce    for  40  dollars ; 
what  part  of  it  must  each  have  ?  What  did  each  oue'e 
share  amount  to  ? 

124.  Three  men  bought  a  lottery  ticket  for  10  dol- 
lars ;  the  first  gave  3  dollars,  the  second  5  dollars,  and 
the  third  2  dollars.     They  drew  a  prize  of  120  dol- 
lars.    What  was  each  man's  share  ? 

125.  Two  men  hired  a  pasture  for  58  dollars  ;  one 
put  in  7  horses,  and  the  other  3  horses  ;  what  ought 
each  to  pay  ? 

126.  Three  men  commenced  trade  together ;  they 
put  in  money  in  the  following  proportion  :  the  first,  3 
dollars,  as  often  as  the  second  put  in  4,  and  as  often  as 
the  third  put  in  5  ;  they  gained  87  dollars  ;  what  was 
each  man's  share  of  the  gain  ? 

127.  Two  men  hired  a  pasture  for  32  dollars  ;  the 
first  put  in  3  sheep  for  4  months,  the  second  put  in  4 
sheep  for  5  months  ;  how  much  ought  each  to  pay  ? 

Note.  Three  sheep  for  4  months  is  the  same  as  1 2 
sheep  for  1  month,  4  sheep  for  5  months  is  the  same 
as  20  sheep  for  1  month.  This  question  is  therefore 
the  same  as  if  1  man  put  in  1 2  sheep,  and  the  other 
20  sheep. 

128.  Two  men,  A  and  B,  traded  in  company ;  A 
put  in  1  dollar  for  4  months,  and  B  2  dollars  for  3 
months,  and  they  gained  ninety  cents  ;  how  many  cents 
must  each  have  ? 

129.  Three  men,  A,  B,  and  C,  traded  in  company  ; 
and  put  in  money  in  the  following  proportions  :  A  put 
in  4  dollars  as  often  as  B  put  in  3,  and  as  often  as  C 
put    in    2  ;   A's   money   was    in   2    months,    B's    3 
months,  and  C's  4  months,  and  they  gained  100  dol- 
lars ;  what  was  each  one's  share  ? 

130.  Two  men,  A  and  B,  traded  in  company ;  A 


138  ARITHMETIC.  [Part  1. 

put  in  2  dollars  as  often  as  B  put  in  3  ;  A's  money 
was  employed  7  months,  and  B's  money  5  months  ; 
they  gained  58  dollars  ;  what  was  each  man's  share 
of  the  gain  ? 

131.  Three  men,  A,  B,  and  C,  traded  in  company, 
and  put  in  money  in  the  following  proportions  :  A  put 
in  2  dollars  as  often  as  B  put  in  4,  and  as  often  as  C 
put  in  6.     B's  money  was  in  twice  as  long  as  C's,  and 
A's  two  times  as  long  as  B's  :  they  gained  88  dollars  ; 
what  was  each  one's  share  of  the  gain  ? 

Note.  Interest  is  a  reward  or  premium  allowed  by  a 
debtor  to  a  creditor  for  the  use  of  money.  The  usual  in- 
terest for  1  year,  and  that  which  is  established  by  law  in 
twenty-two  of  the  States  of  this  Union,  is  6  cents  on  a  dol- 
lar, 6  dollars  on  a  hundred  dollars,  or  in  fine  *-fa  of  the 
sum  whatever  be  the  denomination.  It  is  called  6  per 
cent.,  that  is  6  on  the  hundred,  because  it  is  always  reckoned 
by  the  hundred.  So  3  per  cent.,  4  per  cent.,  &c.,  signify 
iflT'  T$~ff'  &c-'  or  so  muc'l1  on  *ne  hundred.  The  teacher 
can  vary  the  examples  to  illustrate  the  different  rates  per 
cent. 

132.  The  interest  of  1  dollar  being  6  cents  for  1 
year,  what  is  the  interest  of  7  dollars  for  the  same 
time?  What  is  the  interest  of  10  dollars?  Of  15  dol- 
lars ?  Of  20  dollars  ?  Of  30  dollars  ?  Of  50  dollars  ? 
Of  75  dollars  ?  Of  100  dollars  ?  Of  118  dollars  ? 

133.  If  the  interest  of  1  dollar  is  6  cents  for  1  year, 
what  would  it  be  for  2  years  ?    What  would  be  the 
interest  of  8  dollars  for  two  years?    Of  17  dollars? 
Of  43  dollars  ? 

134.  If  the  interest  of  100  dollars  is  6  dollars  for 
a  year,  what  would  be  the  interest  of  50  dollars  for 
the  same  time  ?    Of  2  hundred  ?    Of  3  hundred  ?    Of 
4  hundred  ?    Of  1  hundred  and  50  ?    Of  2  hundred 
and  50? 

135.  If  the  interest  of  100  dollars  is  6  dollars  for 


Sect.  15.]  ARITHMETIC.  -  139 

1  year,  what  would  be  the  interest  of  it  for  6  months  ? 
For  3  months  ?    For  4  months  ?    For  8  months  ?   For 
9  months  ?    For  1  month  ?    For  2  months  ?    For  5 
months?    For  7  months  ?    For  10  months?    For  11 
months  ? 

136.  What  is  the  interest  of  100  and  32  dollars  for 

2  years,  at  6  per  cent.  ? 

137.  What  is  the  interest  of  300  dollars  for  1  year 
and  6  months,  at  6  per  cent.  ? 

138.  What  is  the  interest  of  1  dollar  for  6  months 
at  6  per  cent.?     What  for  2  months?    What  for  1 
month  ?  What  for  3  months  ?   4  months  ?  5  months  ? 
7  months  ?  9  months  ?  1 1  months  ? 

139.  What  is  the  interest  of  57  dollars  for  1  year 
and  7  months,  at  6  per  cent.  ? 

140.  What  is  the  interest  of  200  and  67  dollars  for 
1  year  and  4  months,  at  6  per  cent.  ? 

141.  If  the  interest  of  1  year  is  6  per  cent.,  what 
would  be  the  per  cent,  for  2  years  ?     For  3  years  ? 
For  6  months  ?    For  2  months  ?    For  1  month  ?    For 
4  months  ?    For  5  months  ?    For  7  months  ?    For  8 
months  ?  For  9  months  ? 

142.  If  the  interest  of  2  months  or  60  days  is  1 
per  cent.,  what  would  be  the  per  cent,  for  20  days  ? 
What  for  40  days  ?  What  for  15  days  ?  What  for  45 
days  ?   What  for  12  days  ?  What  for  10  days  ?   What 
for  5  days  ? 

143.  What  is  the  interest  of  100  and  37   dollars 
for  2  years  3  months  and  20  days  ? 

1 44.  A  can  do  a  piece  of  work  in  2  days  ;  how 
much  of  it  can  he  do  in  1  day  ? 

145.  B  can  do  a   piece  of  work  in  4  days  ;  how 
much  of  it  can  he  do  in  1  day  ? 

146.  If  A  can  do  J  of  a  piece  of  work  in  1  day, 
and  B  can  do  |  of  it  in  1  day,  how  much  would  both 
do  in  a  day  ?    How  long  would  it   take  them    both 
together  to  do  the  whole  ? 


140  ARITHMETIC.  [Parti 

147.  If  1  man  can  do  a  piece  of  work  in  2  daye 
and  another  in  3  days,  how  much  of  it  would  each 
do  in  a  day  ?     How  much  would  both  together  do  ? 
How  long  would  it  take  them  both  to  do  the  whole  ? 

148.  A  cistern  has  2  pipes;  the  first  will  fill  it  in 
3   hours,  the  second    in   6   hours  ;    how   much   of  it 
would  each  fill  in  an  hour?     How  much  would  both 
together  fill?    How  long  would  it  take  them  both  to' 
fill  it  ? 

1 49.  A  man  and  his  wife  found  by  experience,  that, 
when  they  were  both  together,  a  bushel  of  meal  would 
last  them  only  2  weeks  ;  but  when  the  man  was  gone, 
it  would  last  his  wife  5  weeks  ;  how  much  of  it  did 
both  together  consume  in  1  week  ?    What  part  did  the 
woman  alone  consume  in  1  week  ?    What  part  did  the 
man  alone  consume  in  1  week  ?     How  long  would  it 
last  the  man  alone  ? 

150.  If  1  man  could  build  a  piece  of  wall  in  5  days, 
and  another  man  could  do  it  in  7  days  ;  how  much  of 
it  would  each  do  in   1   day  ?     How  many  days  would 
it  take  them  both  to  do  it? 

151.  A  cistern  has  3  pipes ;  the  first  would  fill  it  in 
3  hours  ;  the  second  in  6  hours :  the  third  in  4  hours ; 
what  part  of  the  whole  would  each  iill  in  1  hour?  and 
how  long  would  it  take  them  all  to  fill  it,  if  they  were 
all  running  at  once  ? 

152.  A  and  B  together  can  build  a  boat  in  8  days, 
and  with  the  assistance  of  C  they  can  do  it  in  5  days ; 
how  much  of  it  can  A  and  B  build  in  1  day  ?     How 
much  of  it  can  A,  B,  and  C,  build  in  1  day  ?     How 
much  of  it  can  C  ,.build  alone  in  1  day  ?     How  long 
would  it  take  C  to  build  it  alone  ? 

153.  Suppose  I  would  line  8  yards  of  broadcloth 
that  is  1  £  yards  wide,  with  shalloon  that  is  f  of  a  yard 
wide  ;  how  many  yards  of  the  shalloon   will  t  line   1 
yard  of  the  broadcloth  ?     How  many  yards  will  line 
the  whole? 


Sect.  15.]  ARITHMETIC.  141 

154.  If  7  yards  of  cloth  cost  13  dollars,  what  will 

10  yards  cost? 

155.  If  the  wages  of  25  weeks  come  to  75"  dollars, 
what  will  be  the  wages  of  7  weeks  ? 

156.  If  8  tons  of  hay  will  keep  7   horses  three 
months,  how  much  will    keep    12    horses  the   same 
time  ? 

157.  If  a  staff  4  feet  long  cast  a  shadow  6  feet  long, 
what  is  the  length  of  a  pole  that  casts  a  shadow  58 
feet  at  the  same  time  of  day  ? 

158.  If  a  stick  8  feet  long  cast  a  shadow  2  feet  in 
length,  what  is  the  height  of  a  tree  which  casts  a 
shadow  42  feet  at  the  same  time  of  day  ? 

159.  At   6  dollars  per  week,  how  many  months' 
board  can  I  have  for  100  dollars  ? 

1 60.  A  ship  has  sailed  24  miles  in  4  hours ;  how 
long  will  it  take  her  to  sail  150  miles  at  the  same 
rate  ? 

161.  30  men  can  perform  a  piece  of  work  in  20 
days  ;  how  many  men  will  it  take  to  perform  the  same 
work  in  8  days? 

162.  17  men  can  perform  a  piece  of  work  in  25 
days ;   in  how  many  days  would  5  men  perform  the 
same  work  ? 

163.  A  hare  has  76  rods  the  start  of  a  greyhound, 
but  the  greyhound  runs  15  rods  to  10  of  the  hare  ; 
how  many  rods  must  the  greyhound  run  to  overtake 
the  hare  ? 

164.  A  garrison  has  provision  for  8  months,  at  the 
rate  of  15  ounces  per  day ;  how  much  must  be  al- 
lowed per  day,  in  order  that  the  provision  may  last 

1 1  months  ? 

165.  If  8  men  can  build  a  wall  15  rods  in  length  in 
10  days,  how  many  men  will  it  take  to  build  a  wall 
45  rods  in  length  in  5  days  ? 

166.  If  a  quarter  of  wheat  affords   60  ten-penny 
loaves,  how  many  eight-penny  loaves  may  be  obtained 
from  it  ? 


142  ARITHMETIC.  [Par/  1 

167.  Said  Harry  to  Dick,  My  purse  and  money  to- 
gether are  worth  1 6  dollars,  but  the  money  is  worth  7 
times  as  much  as  the  purse  ;  how  much  money  was 
there  in  the  purse  ?    and  what  is  the  value  of  the 
purse  ? 

168.  A  man  being  asked  the   price  of  his  horse, 
answered,  that   his   horse  and  saddle  together  were 
worth  100  dollars,  but  the  horse  was  worth  9  times  as 
much  as  the  saddle  ;  what  was  each  worth  ? 

169.  A  man  having  a  horse,  a  cow,  and  a  sheep, 
was  asked  what  was  the  value  of  each.    He  answered, 
that  the  cow  was  worth  twice  as  much  as  the  sheep, 
and  the  horse  3  times  as  much  as  the  sheep,  and  that 
all  together  were  worth   60  dollars  ;   what  was  the 
value  of  each  ? 

170.  A  man  bought  an  apple,  an   orange,  and  a 
melon,  for  21  cents  ;  for  the  orange  he  gave  twice  as 
much  as  for  the  apple,  and  for  the  melon  he  irave  twice 
as  much  as  for  the  orange  ;  how  much  did  he  give  for 
each  ? 

171.  If  80  dollars'  worth  of  provision   will  serve 
20  men  24  days,  how  many  days  will   100  dollars' 
worth  of  provision  serve  30  men  ? 

172.  There  is  a  pole  £  and  £  under  water,  and  10 
feet  out ;  how  long  is  the  pole  ? 

173.  In  an  orchard  of  fruit  trees,  J  of  them  bear 
apples,  ^  of  them  bear  plums,  £  of  them  pears,  7  of 
them  peaches,  and  3  of  them  cherries ;    how  many 
trees  are  there  in  the  whole,  and  how  many  of  .each 
sort? 

174.  A  farmer  being  asked  how  many  sheep  he 
had,  answered,  that  he  had  them  in  4  pastures ;   in 
the  first  he  had  £  of  his  flock ;  in  the  second  £ ;  in 
the  third  £  ;  and  in  the  fourth  15  ;  how  many  sheep 
had  he? 

175.  A  man  driving  his  geese  to  market,  was  met 
by  another,  who  said,  Good-morrow,  master,  with  jour 


Serf.  15.]  ARITHMETIC.  J  i;j 

hundred  geese ;  says  he,  I  have  not  a  hundred ;  but  if 
I  had  half  as  many  more  as  I  now  have,  and  two  geese 
and  a  half,  I  should  have  a  hundred ;  how  many  had 
he? 

176.  What  number  is  that,  to  which  if  its  half  be 
added  the  sum  will  be  60? 

177.  What  number  is  that,  to  which  if  its  third  be 
added  the  sum  will  be  48  ? 

178.  What  number  is  that,  to  which  if  its  fifth  be 
added  the  sum  will  be*  54? 

179.  What  number  is  that,  to  which  if  its  half  and 
its  third  be  added  the  sum  will  be  55  ? 

180.  A  man  being  asked  his  age,  answered,  that  if 
its  half  and  its  third  were  added  to  it,  the  sum  would 
be  77  ;  what  was  his  age  ? 

181.  What  number  is  that  which   being  increased 
by  its   half,  its  fourth,  and   eighteen   more,  will  be 
doubled  ? 

182.  A  boy  being  asked  his  age,  answered,  that  if 
J  and  |  of  his  age,  and  20  more,  were  added  to  his 
age,  the  sum  would  be  3  times  his  age.    What  was  his 
age? 

183.  A  man  being  asked  how  many  sheep  he  had, 
answered,  that  if  he  had  as  many  more,  J  as  many 
more,  and  2i  sheep,  he  should  have  100.     How  many 
had  he? 


144 


ARITHMETIC!. 


[Part  I. 


TABLES     FOR    REVIEW. 


No.  1. 

7  times   8 
12     "       9 
3     "     11 
6     "       6 
10     "       4 

No.  7. 

8  times   6 
3     "       8 
6     "       7 
4     "     12 
9     "     11 

No.  13. 

5  times   8 
4     "       9 
11     "     10 
7     "       6 
8     "     12 

No.  19. 

8  times    7 
12     "       6 
3     "       6 
9     "     10 
2     "       4 

No.  25. 

4  times    3 
10     "       6 
8     "       9 
7     "       2 
12     "       6 

No.  2. 

2  times    7 
10     "       8 
12     "     11 
4     "       6 
3     "       9 

No.  8. 

6  times    9 
11      "       8 
7     "     10 
2     "     12 
3     "       4 

No.  14. 

12  times    4 
6     "       9 
3     "       8 
6     "     11 
10     "       7 

No.  20. 

9  times    6 
2     "     10 
12     "       7 
6     "       6 
11     "       4 

No.  26. 

8  times    6 
6     "      12 
10     "       9 
11     "       2 
7     "       3 

No.  3. 

6  times  11 
9     "       7 
10     "      10 
4     "       8 
3     "       6 

No.  9. 

11  times  11 

4     "       7 
10     "        5 
8     "       3 
6     "       2 

No.  15. 

9  times    6 
7     "     11 
8     "       4 
6     "       3 
10     "       2 

No.  21. 

2  times    3 
7     "       7 
11      "       0 
4     "     10 
8     "      12 

No.  27. 

12  times    3 

7     "       6 
9     "       4 

6     "       8 
2     "       2 

No.  4. 

3  times    7 
8     "      10 
9     "       9 
6     "       6 
12     "       2 

ISo.  10. 

8  times   8 
9     "      12 
2     "       5 
4     "      11 
6     "       7 

No.  16. 

10  times  12 

7     "       9 
3     "       2 
4     "       6 
8     "     11 

No.  22. 

9  times    8 
6     "      10 
4     "       2 
11     "       7 
12     "     12 

No.  28. 

8  times    2 
6     "       6 
11      "       9 
10     "       3 
6     "       4 

No.  5. 

11  times    3 
12     "       8 
2     "       9 
4     "       4 
6     "     10 

No.  11. 

9  times    3 
10     "     11 
7     "       4 
6     "     12 
2     "       6 

No.  17. 

2  times  11 
3     "     10 

7     "      12 
9     "       6 
8     "       4 

No.  23. 

11  times    5 
9     "       2 
6     "       3 
12     "      10 
7     "       8 

No.  29. 

6  times    2 
6      "       8 
7     "       3 
11     "     12 

9     «       4 

No.  6. 

11  times  11 
9     "       7 
12     "       8 
6     "       4 
11     "       8 

No.  12. 

12  times    9 
10     "     11 
7     "       6 
8     "       8 
6     "       4 

No.  18. 

11  times  12 
9     "       9 

8     "       3 
7     "       4 
6     "       6 

No.  24. 

8  times    9 
12     "     10 
11     "       7 
12     "       6 
6     "       9 

No.  30. 

12  times    7 
9      "      11 
12     "     12 

11     "       6 

7      "       5 

EXPLANATION  OF  THE  TABLES.  —  These  Tables  are  designed  not  for  study, 
but  for  an  accompanying  exercise  to  the  usual  recitation.     By  their  use, 
many  words  will  be  saved,  as  the  teacher  will  not  be  obliged  to  ask  the 
questions  ;  and  hence  more  can  be  accomplished  in  a  given  time.    By  sub- 
stituting and  for  times  and  in.  Addition  can  also  be  reviewed  ;  by  substi- 
tuting /rom,  and  mentioning  the  smaller  number  first,  Subtraction.    In 

Sect.  15.] 


ARITHMETIC. 


143 


TABLES    FOR    REVIEW. 


No.  1. 

11  in  121 
4  "     16 
7  "     49 
5  "     60 
8  "     32 

No.  7. 
10  in    70 
3  "     27 
6  "     48 
2  "     14 
9  "     72 

No.  13. 

3  in     30 
9  "     45 
12  "   108 
8  "     96 

7  "     49 

No.  19. 

6  in     72 

7  "     56 
11  "     77 
4  "     48 
5  "     30 

No.  25. 

2  in  24 

9  "     90 
12  "     72 
7  "     35 
4  "       8 

No.  2. 

3  in    33 
8  "     64 
12  "   120 
6  "     35 
10   "    '20 

No.  8. 

4  in    28 
11  "     55 
10  "     90 
6  "     36 
2  "     12 

No.  14. 

2  in    16 
5  "     50 
12  "     48 
3  "     21 
7  "     70 

No.  2O. 

8  in    80 
11  "     33 
10  "   110 
5  "     25 
3  "       9 

No.  26. 

6  in  66 
4  "  40 
9  "  108 
3  "  27 
8  "  16 

No.  3. 

12  in  144 
5  "     20 
6  "     42 
7  "     28 
3  "     15 

No.  9. 

6  in    54 
2  "     22 
11  "     88 
10  "     50 
9  "     63 

No.  15. 

4  in    36 
9  "     99 
11   "     44 
10  "   100 
7   "     63 

No.  21. 

5  in    40 
2  "     18 
12  «   132 
2  "       6 
9  "     27 

No.  27. 

11  in    66 
6  "     24 
5  "     15 
12  "     60 
7  "     14 

No.  4. 

7  in    42 
4  "     20 
12  "     84 
6  «     72  • 
9  "     36 

No.  1O. 

10  in    60 
11  "     99 
8  "     24 
3  "     12 

6  "     18 

No.  16. 

2  in      8 
11  "     22 
10  "     40 

7  "     77 

3  "     18 

No.  22. 

4  in     32 
6  "     60 
12  "     96 
9  "     18 
3  "     24 

No.  28. 

11  in  132 
8  "  88 
2  "  20 
9  "  54 
7  "  21 

No.  5. 

2  in      4 
12  "     36 
10  «     80 
4  «     12 

7  "     84 

No.  11. 

12  in    24 
10  '•   120 
6  "     55 
8  "     48 
6  "     30 

No.  IT. 

8  in     56 
5  "     10 
4  "     44 
8  "     72 
11  "   110 

No.  23. 

4  in     24 
8  "     40 
3  "       6 
10  "     30 
9  "     81 

No.  29. 

2  in     10 
11  "    121 

12  "  108 
9  "  72 

7  "  84 

No.  6. 

12  in  132 
7  «     63 
10  "   110 
9  "     54 
8  "     88 

No.  12. 

10  in  120 
12  "     96 
6  "     72 
9  "     81 

7  "     28 

No.  18. 

5  in     45 
12  "   144 
7  "     56 
8  "     48 
3  "     36 

No.  24. 

4  in     56 
12  «     60 
5  "     45 
6  "     42 

8   "     64 

No.  30. 

7  in     35 
12  "     48 
9  "     99 

6  "  36 
8  "  24 

many  ways  can  the  tables  be  used  to  advantage.     The  columns  can  be 
added,  or  the  numbers  added  across  the  tables.    The  divisor  of  one 
number  can  be  subtracted  from  the  dividend  of  another,  or  the  dividend 
of  one  number  can  be  divided  by  the  divisor  of  another.     For  example, 
take  Nos.  1  and  7  :  11  from  70,  4  from  27,  10  from  121,  3  from  16  ;  or  11, 
in  70,  4  in  27,  10  in  121,  3  in  16. 

ARITHMETIC. 
PART  II. 

KEY. 

THE  Key  contains  remarks  on  the  principles  employed 
and  illustrations  of  the  manner  of  solving  the  examples  is 
each  section.  All  the  most  difficult  of  the  practical  exam- 
ples are  solved  in  such  a  manner  as  to  show  the  principles 
by  which  they  are  performed. 

Instructors  who  may  never  have  attended  to  fractions 
need  not  be  afraid  to  undertake  to  teach  this  book.  The 
author  flatters  himself  that  the  principles  are  so  illustrated, 
and  the  processes  are  made  so  simple,  that  any  one,  who 
shall  undertake  to  teach  it,  will  find  himself  familiar  with 
fractions  before  he  is  aware  of  it,  although  he  knew  noth- 
ing of  them  before ;  and  that  every  one  will  acquire  a  fa- 
cility in  solving  questions  which  he  never  before  possessed. 

The  reasoning  used  in  performing  these  small  examples 
is  precisely  the  same  as  that  used  upon  large  ones.  And 
when  any  one  finds  a  difficulty  in  solving  a  question,  he  will 
remove  it  much  sooner,  and  much  more  effectually,  by 
taking  a  very  small  example  of  the  same  kind,  and  observ- 
ing how  he  does  it,  than  by  recurring  to  a  rule. 

The  practical  examples  at  the  commencement  of  each 
section  and  article  are  generally  such  as  to  show  the  pupil 
what  the  combination  is,  and  how  he  is  to  perform  it.  Thus 
the  pupil  will  learn  gradually  to  reason  upon  abstract  num- 
bers. In  each  combination,  there  are  a  few  abstract  exam- 
ples without  practical  ones,  to  exercise  the  pupil  in  the 
combinations,  after  he  knows  what  these  combinations  are. 
It  would  be  an  excellent  exercise  for  the  pupil  to  put  these 
into  a  practical  form  when  he  is  reciting.  For  instance, 


Sect.  1.]  KEY.  1  17 

when  the  question  is,  How  many  are  5  and  8  ?  Let  him 
make  a  question  in  this  way :  if  an  orange  cost  5  cents, 
and  an  apple  3  cents,  what  would  they  both  come  to  V 
This  may  be  done  in  all  cases. 

The  examples  are  often  so  arranged,  that  several  depend 
on  each  other,  so  that  the  preceding  explains  the  following 
one.  Sometimes,  also,  in  the  same  example,  there  are  several 
questions  asked,  so  as  to  lead  the  pupil  gradually  from  the 
simple  to  the  more  difficult.  It  would  be  well  for  the  pupil 
to  acquire  the  habit  of  doing  this  for  himself,  when  difficult 
questions  occur. 

The  operations  can  be  illustrated  by  counters,  or  marks 
on  the  blackboard,  according  to  the  necessity  of  the  pupils. 

The  book  may  be  used  in  classes  where  it  is  convenient. 
A  very  useful  mode  of  recitation  is  for  the  instructor  to 
read  the  example  to  the  whole  class,  and  then,  allowing 
sufficient  time  for  them  to  perform  the  question,  call  upon 
some  one  to  answer  it.  In  this  manner  every  pupil  will  be 
obliged  to  perform  the  example,  because  they  do  not  know 
who  is  to  answer  "it.  In  this  way  it  will  be  best  for  them 
to  answer  without  the  book. 

It  will  often  be  well,  to  let  the  elder  pupils  hear  the 
younger.  This  will  be  a  useful  exercise  for  them,  and  an 
assistance  to  the  instructor. 


SECTION  I. 

A.  THIS  section  contains  addition  and  subtraction.  The 
first  example  may  be  solved  by  means  of  beans,  peas,  &c., 
or  by  means  of  the  blackboard.  The  former  method  is 
preferable,  if  the  pupil  be  very  young,  not  only  by  the  ex- 
amples in  the  first  part  of  this  section,  but  by  the  first  ex- 
amples in  all  the  sections. 

B  &  C.  The  articles  B  and  C  contain  the  common  ad- 
dition table  as  far  as  the  first  10  numbers.  In  the  firstj  the 
numbers  are  placed  in  order ;  and  in  the  second,  out  of 
order. 

The  pupil  should  study  these  until  he  can  find  the  an- 
swers readily,  and  then  he  should  commit  the  answers  to 
memory. 


148  KEY.  [Par/  2, 

D.  In  this  article  the  numbers  are  larger  than  in  the 
preceding  ;  and,  in  some  instances,  three  or  more  numbers 
are  added  together.     In  the  abstract  examples,  the  num. 
bers  from  one  to  ten  are  to  be  added  to  the  numbers  from 
ten  to  twenty. 

E.  This  article  contains  subtraction. 

F.  This  article  is  intended  to  make  the  pupil  familiar 
with  adding  the  first  nine  numbers  to  all  •  others.     The  pu- 
pil should  study  it  until  he  can  answer  the  questions  very 
readily. 

G.  In  this  article  all  the  preceding  are  combined  to- 
gether, and  the  numbers  from  1  to  10*  are  added  to  all 
numbers  from  20  to  1 00,  and  subtracted  in  the  same  man- 
ner. 

H.  This  article  contains  practical  questions  which  show 
the  application  of  all  the  preceding  articles. 

6.  37  less  5  are  32,  less  8  are  24,  less  6  (which  he  kept 
himself)  are  18 ;  consequently  he  gave  18  to  the  third  boy. 

After  this  section  use  the  tables  on  pp.  144  and  145  for 
review  of  Addition  and  Subtraction. 


SECTION  II. 

THIS  section  contains  multiplication.  The  pupil  will  see 
no  difference  between  this  and  addition.  It  is  best  that  he 
should  not  at  first,  though  it  may  be  well  to  explain  it  to 
him  after  a  while. 

A.  This  article  contains  practical  questions,  which  the 
pupil  will  readily  answer. 

I.  Three  yards  will  cost  3  times  as  much  as  1  yard. 

N.  B.  Be  careful  to  make  the  pupil  give  a  similar  rea- 
son for  multiplication,  both  in  this  article,  and  elsewhere. 

I 1 .  A  man  will  travel  4  times  as  far  in  4  hours  as  he  will 
in  1  hour. 

*  Figures  are  used  in  the  key,  because  the  instructor  is  supposed 
to  be  acquainted  with  them.  They  are  not  used  in  the  first  part  of 
the  book,  because  the  pupil  would  not  understand  them  so  well  aa 
he  will  the  words. 


Sect.  3.]  KEY.  149 

B.  This  article  contains  the  common  multiplication  ta- 
ble, as  far  as  the  product  of  the  first  ten  numbers.     The 
pupil  should  find  the  answers  once  or  twice  thrdfigh,  untij 
he  can  find  them  readily,  and  then  let  him  commit  them  to 
memory. 

C.  This  article  is  the  same  as  the  preceding,  except 
in  this,  the  numbers  are  out  of  their  natural  order. 

D.  In  this  article  multiplication  is  applied  to  practi- 
cal examples.     They  are  of  the  same  kind  as  those  in  ar- 
ticle A  of  this  section. 

12.  There  are  8  times  as  many  squares  in  8  rows  as  in 
1  row.  8  times  8  are  64. 

23.  There  are  3  times  as  many  pints  in  3  quarts  as  in  1 
quart.  3  times  2  are  6.  And  in  6  pints  there  are  6  times 
4  gills,  or  24  gills. 

35.  In  one  gallon  are  32  gills ;  and  32  times  2  cents  are 
64  cents.  Or,  1  pint  will  cost  8  cents,  and  there  are  8 
pints  in  a  gallon.  8  times  8  are  64. 

38.  They  will  be  2  miles  apart  in  1  hour,  4  miles  in  2 
hours,  &c. 

After  this  section  make  use  of  the  table  on  page  144. 


SECTION  III. 

A.  THIS   section   contains   division.     The    pupil   will 
scarcely  distinguish  it  from  multiplication.     It  is  not  impor- 
tant that  he  should  at  first. 

The  pupil  will  be  able  to  answer  these  questions  by  the 
multiplication  table,  if  he  has  committed  it  to  memory 
thoroughly. 

B.  In  this  article  the  pupil  obtains  the  first  ideas  of  frac- 
tions, and  learns  the  most  important  of  the  terms  which  are 
applied  to  fractions.*      The  pupil  has  already  been  ac- 
customed to  look  upon  a  collection  of  units,  as  forming  a 
number,  or  as  being  itself  a  part  of  another  number.     Ho 
knows,  therefore,  that  one  is  a  part  of  every  number,  and 
that  every  number  is  a  part  of -every  number  larger  than 

*  As  soon  as  the  terms  applied  to  fractions  are  fully  comprehend- 
ed, the  operations  on  them  are  as  simple  as  those  on  whole  numbers. 


150  KEY.  [Part  2 

itself.  As  every  number  may  have  a  variety  of  parts,  it  is 
necessary  to  give  names  to  the  different  parts  in  order  to 
distinguish  them  from  each  other.  The  parts  receive  their 
names,  according  to  the  number  of  parts  which  any  num- 
ber is  divided  into.  If  the  number  is  divided  into  two 
equal  parts,  the  parts  are  called  halves ;  if  it  is  divided  into 
three  equal  parts,  they  are  called  thirds  ;  if  into  four  parts, 
fourths,  &c. ;  and  having  divided  a  number  into  parts,  we 
can  take  as  many  of  the  parts  as  *we  choose.  If  a  number 
be  divided  into  five  equal  parts,  and  three  of  the  parts  be 
taken,  the  fraction  is  called  three  fifths  of  the  number. 
The  name  shows  at  once  into  how  many  parts  the  number 
is  to  be  divided,  and  how  many  parts  are  taken. 

The  examples  in  this  book  are  so  arranged  that  the 
names  will  usually  show  the  pupil  how  the  operation  is  to 
be  performed.  In  this  section,  although  the  pupil  is  taught 
to  divide  numbers  into  various  parts,  he  is  not  taught  to 
notice  any  fractions,  except  those  where  the  numbers  aro 
divided  into  their  simple  units,  which  is  the  most  simple 
kind. 

It  will  be  best  to  use  beans,  pebbles,  &c.,  first. 

20.  Since  1  is  1  third  of  3,  2  must  be  2  thirds  of  3. 

34.  Illustrate  by  grouping  the  marks  or  counters  by 
threes. 

Proceed  in  the  same  manner  with  the  other  divisions. 

This  bein»  one  of  the  most  useful  combinations,  and  one 
but  very  little  understood  by  most  people,  especially  when 
applied  to  large  numbers,  the  pupil  must  be  made  perfectly 
familiar  with  it.  Ask  questions  like  those  in  the  oook  for 
large  numbers,  and  also  some  like  the  following :  What 
part  of  7  is  18  ?  the  answer  will  be  y. 

C.  The  first  ten  figures  are  here  explained.    They  are 
used  as  an  abridged  method  of  writing  numbers,  and  not 
with  any  reference  to  their  use  in  calculating. 

This  article  is  only  a  continuation  of  the  last.  All  the 
numbers  from  1  to  100  are  introduced  into  the  two  articles, 
and  are  divided  by  all  the  numbers  from  1  to  10  ;  except 
that  some  of  the  largest  are  not  divided  by  some  of  the 
smallest. 

D.  These  examples,  which  are  similar  to  those  in  article 
A  of  this  section,  are  solved  in  the  same  manner. 


Sect.  4.]  KEY.  .  151 

5.  It  would  take  as  many  hours,  as  8  miles  are  contained 
in  10  miles.  3  hours  and  J  of  an  hour. 

20.  They  cost  as  many  cents  as  there  are  3  apples  in  80 
apples;  that  is,  10  cents. 

26.  The  whole  loss  was  35  dollars,  which  was  7  dollars 
apiece. 

After  this  section  use  the  table  on  page  145. 


SECTION  IV. 

A.  THIS  article  contains  multiplication  simply.     It  is 
repeating  a  number  a  certain  number  of  times  and  a  part 
of  another  time. 

14.  6  times  5  are  30,  and  £  of  5  are  3,  which  added  to 
30  make  33. 

B.  In  this  article  the  pupil  is  taught  to  change  a  certain 
number  of  twos  into  threes,  threes  into  fives,  &c.     This  ar- 
ticle combines  all  the  preceding  operations. 

24.  4  cords  of  wood  will  cost  28  dollars,  and  ^  of  a  cord 
will  cost  2  dollars,  which  makes  30  dollars.  30  dollars  will 
buy  3  hundred-weight  of  sugar  and  f  of  another  hundred- 
weight. 

C.  2.  The  two  lemons  come  to  8  cents,  and  8  cents  will 
buy  4  apples,  at  2  cents  apiece. 

This  is  usually  called  Barter.  The  general  principle  is  to 
find  what  the  article  will  come  to,  whose  price  and  quan- 
tity are  given,  and  then  to  find  how  much  of  the  other  ar- 
ticle that  money  will  buy. 

6.  If  2  apples-  cost  4  cents,  1  will  cost  2  cents,  and  4  will 
cost  8  cents. 

22.  Find  how  many  times  2  pears  are  contained  in  20 
pears,  which  is  10  times.     10  times  3  cents  are  30  cents. 
Or,  first  find  what  20  pears  would  come  to  at  3   centa 
apiece ;  and  since  it -is  2  for  3  cents,  instead  of  1  for  3 ,3ents 
the  price  will  be  half  as  much. 

23.  See  how  many  times  you  can  have  5  cents  in  80 
cents,  and  you  can  buy  so  many  times  3  eggs 

26.  5  dollars  apiece. 


152  .        &EY.  [Par*  2 

SECTION  V. 

IN  this  sectiou  the  principle  of  fractions  is  applied  tc 
larger  numbers,  but  sucn  as  are  divisible  into  the  parts  pro- 
posed to  be  taken.  The  pupil,  who  is  familiar  with  what 
precedes,  will  easily  understand  the  examples  in  this  sec- 
tion. They  require  nothing  but  division  and  multiplication. 

A.  Let. the  pupil  explain  each  example  in  the  following 
manner.    What  is  1  sixth  of  18  ?     Ans. :  3.     Why  ?    Be- 
cause 6  times  3  are  18 ;  therefore  if  you  divide  18  into  6 
equal  parts,  one  of  the  parts  will  be  3. 

The. pupil  will  be  very  likely  to  say  3  is  the  6th  part  of 
18,  because  3  times  6  are  18.  Be  careful  to  make  nun  saj* 
it  the  other  way,  viz.,  6  times  3  are  18. 

B.  11.  1  of  20  is  4 ;  J  are  7  times  4,  or  28  ;  and  28  is  4 
times  6,  and  |  of  6. 

C.  3.  1  half  of  10  is  5,  £  of  ten  are  4  ;  5  and  4  are  9. 
He  gave  away  9,  and  had  1  left. 

5.  2  yards  will  cost  1  half  of  what  4  cost ;  or  6  dollars. 

7.  2  is  J  of  3  ;  therefore  2  oranges  will  cost  £  of  what  3 
cost.     J  of  18  cents  are  12  cents. 

8.  £  of  25  are  20.     The  ten  apples  cost  20  cents,  which 
was  2  cents  apiece. 

13.  3  is  f  of  4.  f  of  12  dollars  are  9  dollars.  Or  4  yards 
at  12  dollars  is  3  dollars  a  yard,  and  9  dollars  for  3  yards 

15.  Since  1  is  J  of  3,  7  is  J  of  3.  J  of  15  cents  are  35 
cents.  Or,  3  oranges  at  15  cents,  is  5  cents  apiece :  7 
times  5  cents  are  35  cents. 

Note.  In  questions  of  this  kind  it  is  generally  the  sim- 
plest way  to  find  wha*  1  article  will  cost,  then  it  may  easily 
be  told  how  much  anj  number  will  cost. 


SECTION  VI. 

A.  4.  2  halves  of  any  number  make  the  whole  number 
Therefore  2  is  1  half  of  2  times  2 ;  or  4.  It  is  \  of  4  times 
2,  or  8. 

Let  the  pupil  answer  these  questions  in  the  following 
manner :  4  is  £  of  3  times  4  ;  3  times  4  are  12.  .  5  is  |  of 
7  times  5  ;  7  tunes  5  are  35. 


Sect.  6.]  KEY.  153 

B.  2.  4  is  2  times  2. 

16.  2  thirds  of  any  number  is  twice  as  mnjh  as  J  of  the 
same  number.  If  4  is  $  of  some  number,  then  1  half  of  4 
or  2  is  4  of  that  number ;  2  is  £  of  6  ;  therefore  4  is  f  of  6. 

23.  ft  is  evident  that  \  of  a  pound  will  cost  only  J-  of 
what  ^  will  cost.  If  f  cost  6  cents,  \  will  cost  2  cents,  and 
the  whole  pound  14  cents. 

26.  It  will  probably  be  perceived  by  this  time,  that  f  of 
a  number  being' given,  it  is  necessary  to  find  ^,  and  then  the 
number  is  easily  found ;  4  being  |,  2  is  ^,  and  2  is  ^  of 
14. 

45.  24  being  f ,  \  of  24  or  3  will  be  £  ;  3  is  £  of  27. 

C.  6.  20  being  ^,  5  is  ^,  and  5  is  \  of  35 ;  and  35  is  5 
times  6,  ancj  |  of  6. 

D.  4.  18  is  3  times  6,  and  6  is  £  of  4  times  6,  or  24. 
Ans.  24  dollars/ 

6.  54  is  |  of  48  ;   12  yards  at  48  dollars  is  4  dollars  a 
yard.     He  gained  6  dollars. 

7.  10  feet  is  f  of  15  feet. 

8.  If  f  are  under  water,  there  must  be  J  out  of  the 
water.     4  is  J  of  12. 

10.  |  and  |  are  f.  ,-f-  bear  cherries  and  peaches;  con- 
sequently, the  10  which  bear  plums  must  be  the  other  ^. 
10  is  f  of  35.  10  bear  peaches,  and  15  bear  cherries. 

Miscellaneous  Examples. 

6.  The  greyhound  gains  upon  the  fox  4  rods  in  a  minute. 
It  will  take  him  20  minutes  to  gain  80  rods. 

8.  |  of  24.     Or  you  may  say,  1  sheep  would  cost  3  dol- 
lars, and  3  sheep  9  dollars. 

9.  30  horses  will  eat  10  times  as  much  as  3  horses. 

12.  5  dollars  for  1  week,  20  dollars  for  a  month,  and  25 
dollars  for  five  weeks. 

14.  It  would  take  them  5  times  as  long  to  eat  40  bush- 
els, as  it  would  to  eat  8  bushels. 

15.  4  horses  would  eat  4  bushels  in  3  days,  and  it  would 
take  them  9  times  as  long  to  eat  36  bushels.   Ans.  27  days. 

16.  If  2  men  spend  12  dollars  in  1  week,  1  man  will 
pend  6  dollars  in  1  week,  and  30  dollars  in  5  weeks,  and 
5  men  would  spend  3  times  as  much,  or  90  dollars. 

17.  The  shadow  of  the  staff  is  J  of  the  length  of  the 


154  KEY.  IP  art  2, 

staff,  therefore  the  shadow  of  the  pole  must  be  J  the  length 
of  the  pole.  18  feet  is  f  of  27  feet. 

20.  It  would  take  2  men  3  times  as  long  to  do  it  as  it 
would  6  men. 

23.  8  men  would  do  a  piece  of  work  1  half  as  large  in  2 
days,  and  it  would  take  2  men  4  times  as  long  to  do  it,  or 
8  days. 

28.  He  must  sell  it  for  56  dollars  in  order  to  gain  16 
dollars.     56  dollars  is  7  dollars  per  barrel. 

29.  It  cost  him  35  dollars,  and  he  must  sell  it  for  45  to 
gain  10  dollars  ;  45  dollars  is  9  dollars  a  firkin. 

30.  Ans.  56  cents.     See  Section  VI. 

33.  If  it  would  last  3   men   10  months,  it  would  last  1 
man  30  months,  and  5  men  6  months. 

34.  There  are  8  times  5  in  40  ;  and  since  the  other  would 
build  as  many  times  9  as  the  first  does  5^  he  would  build 
8  times  9  or  72  rods. 


SECTION  VH. 

A.  13.  i  of  20  is  4,  |  are  16  ;  16  being  f ,  2  is  } ;  2  is  4 
of  14,  and  16  is  f  of  14. 

16.  f  of  28  are  12 ;  12  is  2  times  6,  and  6  is  J  of  48  (12 
is  |  of  48),  and  48  is  6  times  seven  and  ^  of  7. 

B.  1.  $  of  15  are  12  ;  12  is  6  times  2 ;  2  is  ^  of  20  (12 
is  T%  of  20) ;  J  of  21  is  7 ;  20  is  2  times  7  and  |  of  7. 

2.  f  of  18  are  24 ;  24  is  f  of  27 ;  \  of  35  is  5 ;  27  is  5 
times  5  and  £  of  5. 

C.  This    article   contains  the   multiplication    table,   in 
which  the  numbers  from  10  to  20  are  multiplied  by  the  ten 
first  numbers. 


SECTION  VIII. 

A.  1.  In  one  there  are  2  halves;  in  2  there  are  twice  as 
many  halves,  that  is  4  halves,  or  f . 

2.  In  one  there  are  2  halves,  in  3  there  are  three  times 
two  halves,  that  is,  6  halves,  or  -J. 

3.  In  one  there   are  3  thirds,  in   2  there  are  twice  3 
'birds,  that  is,  6  thirds,  or  }. 


Sect.  9.]  KEY.  L55 

15.  Draw  two  linos  on  the  board  thus  -  Each  line 

call  one  ;  divide  the  upper  line  into  two  equal  parts,  each 
part  is  one  half  of  one,  or  one  half ;  divide  the  lower  line 
in  the  same  way,  and  then 'call  for  the  answer.  The  form 
of  the  question  may  be  varied  by  asking  how  many  half 
apples  there  are  in  two  apples. 

3  7.  In  5  and  2  thirds,  how  many  thirds  ?  Draw  five 
parallel  equal  lines,  divide  them  into  thirds,  and  draw 
another  parallel  with  the  others  and  two  thirds  as  long,  let 
Jhe  scholar  count  the  thirds.  In  all  these  cases  of  illustra- 
tion on  the  board,  it  is  presumed  that  the  scholar  will  soon 
take  the  hint,  and  devise  illustrations  for  himself  at  his  seat. 


SECTION  EX. 

A.  2.  J  signifies  that   1  thing  is  divided  into  3  equal 
parts,  and  1  part  taken.     Therefore  2  times  1  third  is  2 
parts,  or  f . 

6.   7  times  J  is  £,  or  2j. 

B.  4.  4  times  2  are  8,  and  4  times  1  half  are  4  halves, 
or  2,  which  added  to  8  make  10. 

18.  4  times  3  are  12,  and  4  times  J  are  ^  or  three 
whole  ones,  which  added  to  12  make  15. 

40.  10  barrels  at  3  dollars  and  |  a  barrel ;  10  barrels 
at  3  dollars,  would  be  30  dollars,  then  10  times  |  is  ^-,  or 
8  and  £  of  a  dollar.  Ans.  38f  dollars. 

C.  2.  |  to  each  would  be  3  times  J,  or  J,  which  are  2£ 
oranges. 

3.  if-  or  2  bushels. 

4.  7  times  f  are  ^ ,  or  5^  gallons. 

5.  8  yards  and  J  or  two  yards,  that  is  10  yards. 

6.  4  times  2  are  8,  and  4  times  f  are  ^2,  or  2-|,  which 
added  to  8  make  lOf  bushels. 

12.  It  would  take  1  man  3  times  as  long  as  it  would  3 
men.  Ans.  13£  days. 

14.  3  men  would  build  3  times  as  much  as  1  man ;  an<3 
in  4  days  they  would  build  4  times  as  much  as  in  1  day. 
Ans.  38|  rods. 


156  KEY.  [Par/  2. 

SECTION  X. 

A.  21.  J  of  1  is  J.     J  of  2  is  2  times  as  much,  or   J. 

(of  4  is  4,  or  1J.  J  of  5  is  f ,  or  1  j.  J  of  6  is  f,'or  2 
of  7  is  I,  or  2$. 

27.  £  of  1  is  \.  \  of  2  is  f  t  of  3  is  f.  }  of  7  is  £,or  If. 

This  manner  of  reasoning  may  be  applied  to  any  num- 
ber. To  find  \  of  38 :  it  is  ^,  for  |  of  38  is  38  times  as 
much  as  ^  of  1,  and  i  of  1  is  ^  consequently  ^  of  38  is  4j£, 
and  $j£  is  5$. 

40.  To  find  |  of  a  number,  J  mast  be  found  first,  and 
then  J  will  be  2  times  as  much. 

74.  ^  of  50  is  ^,  or  5$ ;  £  is  4  times  as  much  ;  4  times 
5  are  20,  4  times  |  are  ^,  or  2J,  which  added  to  20  make 
22f 

Note.  The  manner  employed  in  example  40th  is  best 
for  small  numbers,  and  that  in  the  74th  for  large  numbers. 

B.  -2.  Ans.  If  apiece. 

3.  £  of  3  is  | ;  |  of  a  bushel  apiece. 

4.  I  of  7  is  4^  ;  he  gave  away  4-^,  and  kept  2£. 

7.  |  of  7  is  J,  or  if;    f  of  a  dollar  is  f  of  100  cents, 
which  is  40  cents.     Ans.  1  dollar  and  40  cents  a  bushel. 

8.  J  of  8  is  1*.     f  of  100  is33f 

9.  If  3  bushels  cost  8  dollars,  1  bushel  will  cost  2  dollars 
and  J,  and  2  bushels  will  cost  5J  dollars. 

13.  If  7  pounds  cost  40  cents,  1  will  cost  5-f  cents ;  10 
pounds  will  cost  57^  cents. 

16.  1  pipe  would  empty  it  in  6  hours,  and  7  pipes  would 
empty  it  in  \  of  6  hours,  or  $  of  1  hour,  which  is  $  of  60 
minutes ;  $  of  60  minutes  is  51$  minutes. 


SECTION    XI. 

A.  2.  2  halves  of  a  number  make  the  number,  conse- 
quently 1  and  1  half  is  the  half  of  2  times  1  and  1  half, 
which  is  3. 

15.  4f  is  £  of  5  times  4  and  f,  which  is  22^. 

B.  4.  5  is  3  times  J  of  5,  which  is  f ,  or  If. 

30.  If  8  is  }  of  some  number,  i  of  8  is  \  of  the  same 


Sect.  12.]  KEY.  .       157 

number.  J  of  8  is  2f ,  2f  is  J  of  4  times  2|  which  is  lOf  • 
therefore  8  is  J  of  lOf. 

52.  If  f  of  a  ton  cost  23  dollars,  J  of  a  ton  must  be  £  of 
23,  that  is  4|  dollars,  and  the  whole  would  cost  9  times  as 
much,  that  is,  41-J. 

69.  1  of  65  is  74 :  74  is  \  of  5  times  7£,  which  is  364. 

65  is  |  of  acf 

C.  4.  37  is  |  of  32f,  which  taken  from  3f7  leaves  4f 
Ans.  4J  dollars. 

5.  7  feet  must  be  -|  of  the  whole  pole. 

6.  If  he  lost  -|,  he  must  have  sold  it  for  -J  of  what  it  cost. 
47  is  J  of  6  Of.     Ans.  60  dollars  and  42f  cents. 

Miscellaneous  Examples. 

1.  The  shadow 'of  the  staff  is  4  of  the  length  of  the  staff; 
therefore  the  shadow  of  the  pole  is  £  of  the  length  of  the 
pole.     67  is  |  of  83|.     Ans.  83|  feet. 

2.  9  gallons  remain  in  the  cistern  in  1  hour.     It  will  be 
filled  in  10  hours  and  J ;  J  of  60  minutes  are  46  minutes 
and  f ;  f  of  60  seconds  are  40  seconds.     Ans.  10  hours,  46 
minutes,  40  seconds. 

10.  Find  |  of  33,  and  subtract  it  from  17.     Ans.  8$. 

11.  It  will  take  3  times  10  yards. 

13.  5  is  |  of  3 ;  it  will  take  {  as  muefy.  Or,  7  yards,  5 
quarters  wide,  are  equal  to  35  yards  1  quarter  wide,  which 
is  equal  to  llf  yards  that  is  3  quarters  wide. 

15.  |  of  37  dollars. 


SECTION   XII. 

The  examples  in  this  section  are  performed  in  precisely 
the  same  manner  as  those  in  the  sections  to  which  they  re- 
fer. All  the  difficulty  consists  in  comprehending,  that 
fractions  expressed  in  figures  signify  the  same  thing  as 
when  .expressed  in  words.  Make  the  pupil  express  them 
in  words,  and  all  the  difficulty  will  vanish.  Let  particular 
attention  be  paid  to  the  explanation  of  fractions  given  in 
the  section. 

VIII.  A.  6.  In  7  how  many  J  ?  •  expressed  in  words,  is 
»n  7  how  many  sixths?  Ans.  y. 


158  KEY.  [Part  2 

14.  Reduce  8^  to  an  improper  fraction  ;  that  is,  in  8 
and  3  tenths,  how  many  tenths.  Ans.  |$. 

B.  8.  y  are  how  many  times  1  ?  That  k,  in  23  sev 
enths  how  many  whole  ones  ?  .4ns.  3£. 

IX.  B.  3.  How  much  is  5  times  64  V  That  is,  how  much 
is  5  times  6  and  4  sevenths  ?  Ans.  324. 

V.  &  X.  15.  What  is  4  of  27  ?     ^That  is,'  what  is   5 
eighths  of  27  V     Ans.  16J. 

VI.  &  XI.  A.  8.  74  is  £  of  what  number  ?     That  is,  7 
and  6  sevenths  is  1  eignth  of  what  number  ?     Ans.  62$. 

SECTION  XIII. 

THE  operations  in  this  section  are  the  reducing  of  frac- 
tions to  a  common  denominator,  and  the  addition  and  sub- 
traction of  fractions.  The  examples  will  generally  show 
what  is  to  be  done,  and  how  it  is  to  be  done. 

4.    It  will  readily  be  seen  that  \  and  \  are  J. 

25.  In  the  fourth  square  of  the  second  row,  it  will  be 
seen  that  1  half  is  •£  ;  and  in  the  second  square  of  the 
fourth  row,  \  is  f  ,  both  together  make  $  and  \  make  -J. 

27.  J  is  the  same  as  -J. 

When  these  questions  are  performed  in  the  mind,  the 
pupil  will  explain  them  as  follows.  He  will  probably  do  it 
without  assistance.  Twenty  twentieths  make  one  whole 
one.  \  of  20  is  5,  and  f  of  20  is  8,  and  ^  of  20  is  2  ; 
therefore  \  is  A,  f  is  ^,  and  ^  is  ^.  All  the  examples 
should  be  explained  in  the  same  manner. 

45.  One  whole  one  is  {?  J,  one  eighth  of  £|  is  -fa.  f  is  3 
times  as  much,  which  is  |^. 

51.  1  half  is  |,  and  J  is  f,  which  added  together  make,  f. 
is  ^,  T8^  is  ^,  J  is  -^,  which  added  together  make 


is  T8^,  |  is  T9^,  which  added  together  make  ^-J  ; 

r  take  A,  and  there  remains  ^|,  or  1. 

It  will  be  easily  perceived  that  these  examples  do 
not  differ  from  those  in  the  first  part  of  the  section,  except 
in  the  language  used.  They  must  be  reduced  to  a  common 
denominator,  and  then  they  may  be  added  and  subtracted 
as  easily  as  whole  numbers.  £  is  ^,  and  $  is  ^,  and  both 
together  make  ||  or  1^. 


Sect.  14.]  KEY.  159 

86  4  is  £,  and  £  is  j.  If  $  be  taken  from  J,  there  re- 
mains |. 

B.  This  article  contains  only  a  practical  application  of 
"the  preceding. 

3.  This  example  and  some  of  the  following  contain 
mixed  numbers,  but  they  are  quite  as  easy  as  the  others. 
The  whole  numbers  may  be  added  separately,  and  the 
fractions  reduced  to  a  common  denominator,  and  then  add' 
ed  as  in  other  cases,  and  afterwards  joined  to  the  whole 
numbers.  6  and  2  are  8  ;  1  half  and  J  are  £,  making  in 
the  whole  8|  bushels. 

C.  It  is  difficult  to  find  examples  which  will  aptly  illus- 
trate this  operation.     It  can  be  done  more  conveniently  by 
the  instructor.     Whenever  a  fraction  occurs,  which  may 
be  reduced  to  lower  terms,  if  it  be  suggested  to  the  pupil, 
he  will  readily  perceive  it  and  do  it. 


SECTION  XIV. 

A.  THIS  section  contains  the  division  of  fractions  by 
whole  numbers,  and  the  multiplication  of  one  fraction  by 
another.  Though  these  operations  sometimes  appear  to  bo 
division,  and  sometimes  multiplication,  yet  there  is  actually 
no  difference  in  the  operations. 

1  and  2.    ^  of  ^  is  ^  of  the  whole. 

3  and  4.     \  of  J  is  J-. 

33.  Since  |^  of  a  share  signify  3  parts  of  a  share,  it  is 
evident  that  J  of  the  three  parts  is  1  part,  that  is,  \. 

39.  ^  signify  9  pieces  or  parts,  and  it  is  evident  that  \ 
of  9  parts  is  3  parts,  that  is,  f  . 

43.  We  cannot  take  J  of  '5  pieces,  therefore  we  must 
take  J  of  fa  which  is  ^,  and  f  is  5  times  as  much  as  fa 
therefore  4  of  f  is  A-. 

78.  8f  is  \2,  1  o£  ^  is  f. 

79.  8^-  is  ^  ,  $  of  }  is  ^,  consequently,  \  of  ^P  is  -|$,  or 


86.  We  may  say  J  of  8-|  is  2,  and  2|  over,  then  2£  is 
V*,  and  J  of  %2  is  ff  ,  hence  J  of  8|  is  2ff 

B.  4.  It  would  take  1  man  4  times  9f,  or  37^  days,  and 
7  men  would  do  it  in  \  of  that  time,  that  is,  ;n  5^-J  days. 


160  KEY.  [Part  2. 

SECTION  XV. 

A.  THIS  section  contains  the  divisions  of  whole  numbers 
by  fractions,  and  fractions  by  fractions. 

1.  Since  there  are  J  in  2,  it  is  evident  that  he  could  give 
them  to  6  boys  if  he  gave  them  J  apiece ;  but,  if  he  gave 
them  £  apiece,  he  could  give  them  to  only  one  half  as 
many,  or  3  boys. 

5.  If  £  of  a  barrel  would  last  them  one  month,  it  is 
evident  that  4  barrels  would  last  20  months ;  but,  since  it 
takes  |  of  a  barrel,  it  will  last  them  but  one  half  as  long, 
or  10  months. 

7.  6f  is  V.  If  J  of  a  bushel  would  last  a  week,  6f 
bushals  would  last  27  weeks ;  but,  since  it  takes  },  it  wiU 
Jast  only  4  of  the  time,  or  9  weeks. 

13.  If  he  had  given  i  of  a  bushel  apiece,  he  might  have 
given  it  to  17  persons ;  out,  since  he  gave  3  halves  apiece, 
ne  could  give  it  to  only  J  of  that  number,  that  is,  to  5 
persons,  and  he  would  have  1  bushel  left,  which  would  be 
§  of  enough  for  another. 

31.  4^  is  ^U,  and  9$  is  *^.  Now  £  is  contained  in  ^ 
48  times,  ana  ^  is  contained  only  ^  part  as  many  times, 
consequently  only  2^6r  or  2^. 

B.  1.  ^  is  T^j- ;  consequently  5  pounds  can  be  bought  for 
J  of  a  dollar. 

3.  |  is  T^,  and  J  is  ^.  If  he  had  given  only  ^  apiece, 
he  could  have  given  it  to  9  persons ;  but  since  he  gave  ^, 
he  could  give  it  to  only  1  half  as  many,  or  4£  persons. 

5.  $  is  ^T,  and  §  is  \\.  If  a  pound  had  cost  -^  of  a  dol- 
lar, 14  pounds  could  be  bought  for  ^i  of  a  dollar ;  but 
since  it  costs  ^T,  only  £  as  many  can  be  bought ;  that  is, 
4§  pounds. 

12.  -|  is  y^,  and  |  is  ^-| ;  ^  is  contained  in  ^£15  times 
fut  ^  is  contained  only  J  as  many  times ;  that  is,  3|  times 

Miscellaneous  -Examples. 

5.  |  of  a  penny  is  J  of  4  farthings.     Ans.  2|  farthings. 

7.  4  of  4  quarters  is  *2  quarters  and  f  of  a  quarter ;  $ 
oi  a  quarter  is  $  of  4  nails,  which  is  1|  nails.  Ans.  2 
quarters,  1|  nails. 

14.  |  of  24  hours  is  14  hours  and  f  of  an  hour ;  f  of  60 
minutes  is  24  minutes.     Ans.  14  hours,  24  minutes. 


Sect.  15.]  KEY.  161 

28.  There  being  4  farthings  in  a  penny,  1  farthing  is  \ 
part  of  a  penny. 

31.  1  penny  is  ^  of  a  shilling,  because  there  are  12 
pence  in  a  shilling. 

48.  1  farthing  is  -fa  of  one  shilling. 

49.  2  farthings  is  ^,  or  ^  of  a  shilling.     5  farthings  ifl 
•fa  of  a  shilling. 

51.  1  penny  is  -^-^  of  1  pound.     7  pence  is  ^fa  of  £l. 
59.  3s.  5d.  is  41  pence,  which  is  ^^  of  £l. 
75.  1  nail  is  -^  of  a  yard.     5  nans  is  A  of  a  yard. 
100.  At  the  end  of  1  hour  they  would  be  7  and  |  miles 
apart.     In  7  hours,  7  times  7£,  which  is  54|  miles.  . 

121.  This  is  the  principle  of  fellowship  ;  3  shillings  were 
paid  ;  one  paid  4,  the  other  f . 

122.  One  paid  f ,  the  other  |. 

126.  3  and  4  and  5  are  12.  The  first  put  in  ^  ;  the 
second  ^  ;  the  third  -fy. 

129.  4  dollars  for  2  months  is  the  same  as  8  dollars  for  1 
month ;  3  dollars  for  3  months  is  the  same  as  9  dollars  for 
1  month  ;  and  2  dollars  for  4  months  is  the  same  as  8  dol- 
lars for  1  month. 

131.  A's  money  was  in  4  times  as  long  as  C's.  It  is  the 
same  as  if  A  had  put  in  8  dollars  for  the  same  time,  and  B 
8  dollars  for  the  same  time.  A  must  have  ^,  B  7^,  and 
C  ^,  of  88  dollars. 

139.  The  interest  of  50  dollars  for  1  year  and  6  months 
is  4  dollars  and  50  cents,  and  for  1   month  it  is  25  cents. 
The  interest  of  7  dollars  for  19  months  (a  dollar  is  ^  of  a 
cent  a  month)  is  66^  cents.     The  whole  amounts  to  5  dol- 
lars and  41^  cents. 

140.  The  interest  of  200  dollars  for  1J  years  is  16  dol- 
lars.    The  interest  of  67   dollars  is  67  cents  for  every  2 
months,  for  16  months  it  will  be  8  times  67  cents,  which 
are  5  dollars  and  36  cents.     The  whole  interest  is  21  dol- 
lars and  36  cents. 

143.  The  interest  of  100  dollars  for  2£  years  is  13  dol- 
lars and  50  cents.  The  interest  of  100  dollars  for  60  days 
would  be  1  dollar,  the  interest  for  20  days  will  be  J  of  a 
dollar,  or  33  J  cents.  The  interest  of  1  dollar  for  2J  years 
is  13^  cents ;  for  10  dollars  the  interest  would  be  1  dollar 
and  35  cents,  and  for  30  dollars,  4  dollars  and  5  cents. 
The  interest  of  7  dollars  for  2J  years  is  7  times  13  i  cents, 
11 


162  KEY.  [Par/ 2 

or  94 £  cents.  The  interest  of  37  dollars  for  60  days  would 
be  37  cents,  and  for  20  days  £  of  37  cents,  or  12J  cents. 
The  whole  interest  is  18  dollars  and  95  J  cents. 

146.  They  would  both  together  do  J  of  the  work  in  1 
day,  and  it  would  take  them  J  of  a  day  to  do  the  other  J 
Ans.  1J  day. 

149.  They  both  together  consume  \  of  a  bushel  in  a 
wees,  but  the  woman  alone  consumes  only  \  of  a  bushel  in 
a  week.  That  is,  they  both  together  consume  A  in  a  week, 
but  the  woman  alone  only  A  ;  consequently,  the  man  alone 
would  consume  ^ ;  and  a  oushel  would  last  him  3  J  weeks. 

152.  A  and  B  can  build  \  of  it  in  1  day ;  A,  B,  and  C, 


can  build  \  of  it  in  1  day ;  the  difference  between  \  and  4 
is  ^  ;  therefore  C  can  build  A  of  it  in  1  day  ;  and  it  would 
take  him  13J  days  to  build  it  alone. 

164.  Find  how  much  they  might  eat  in  a  day,  in  order 
to  make  it  last  1  month,  and  then  it  will  be  easy  to  find 
how  much  they  may  eat  in  a  day,  to  make  it  last  1 1  months. 

167.  The  money  is  7  parts  of  the  whole,  and  the  purse 
one  part ;  consequently  the  money  is  £  and  the  purse  \  of 
16. 

170.  He  gave  one  part  for  the  apple,  2  parts  for  the 
orange,  and  4  parts  for  the  melon.  These  make  7  parts. 
The  apple  3  cents,  the  orange  6  cents,  and  the  melon  12 
cents. 

1 75.  If  to  a  number  half  of  itself  be  added,  the  sum  is 
|  of  that  number ;  hence  subtract  2^  from  100,  and  the 
remainder  is  £  of  the  number  of  geese  that  he  had. 

180.  This  must  be  reduced  to  6ths.     1  half  is  $,  and  J 
is  f ,  and  the  number  itself  is  £.     If  therefore  to  the  whole 
number  its  half  and  its  third  be  added,  the  sum  will  be 
!£• ;  hence,  77  is  ^  of  the  number. 

181.  i  is  £ ;  therefore  if  to  a  number  \  and  \  of  itself 
be  added,  the  whole  number  will  be  J ;  but  when  1 8  more 
is  added  to  J,  the  first  number  is  doubled ;  that  is,  the 
number  is  £  of  the  first  number ;  therefore  18  is  J  of  the 
Dumber. 


ARITHMETIC. 


PART  III. 


WRITTEN  ARITHMETIC. 


Notation  and  Numeration. 

!•     Instead  of  writing  the  names  of  numbers,  it  is  usual 
to  express  them  by  particular  characters,  called  figures. 


One 

is  written    1 

Two 

it 

44 

2 

Three 

44 

44 

3 

Four 

44 

U 

4 

Five 

U 

44 

5 

Six 

44 

U 

6 

Seven 

44 

11 

7 

Eight 

44 

44 

8 

Nine 

U 

a 

9 

Ten 

44 

44 

10 

Eleven 

44 

44 

11 

Twelve 

44 

44 

12 

Thirteen 

M 

44 

13 

Fourteen 

U 

U 

14 

Fifteen 

44 

44 

15 

Sixteen 

« 

U 

16 

Seventeen 

44 

44 

17 

Eighteen 

tl 

44 

18 

Nineteen 

44 

44 

19 

Twenty 

tt 

44 

20 

Twenty-one 

11 

44 

21 

Twenty-two 

44 

44 

22 

Twenty-three 
Twenty-four 

44 
44 

U 

23 
24 

Twenty-five 

U 

14 

25 

1 

Twenjy-six 

is  written  26 

2 

Twenty-seven 

u 

tt 

27 

3 

Twenty-eight 

Li 

tt 

28 

4 

Twenty-nine 

11 

tl 

29 

5 

Thirty 

44 

U 

30 

6 

Thirty-one 

U 

11 

31 

7 

Forty 

It 

It 

40 

8 

Fifty 

tl 

It 

50 

9 

Sixty 

11 

tt 

60 

10 

Seventy 

tl 

It 

70 

11 

Eighty 

tt 

tl 

80 

12 

Ninety 

44 

" 

90 

13 

One  hundred 

44 

tl 

100 

14 

One  hundred 

15 

and  one 

a 

tl 

101 

16 

One  hundred 

17 

and  two 

u 

11 

102 

18 

One  hundred 

19 

and  ten 

u 

11 

110 

20 

One  hundred 

21 

and  twenty 

u 

u 

120 

22 

Two  hundred 

tt 

u 

200 

23 

Three  hundred 

tt 

11 

300 

24 

One  thousand 

U 

« 

1000 

25 

Ten  thousand 

tt 

11 

10,000 

164 


ARITHMETIC. 


2.  When  several  figures  stand  side  by  side,  the  value  of 
each  figure  depends  on  its  place  as  counted  from  the  right 
Awards  the  left.  A  figure  standing  in  the  first  place 
signifies  so  many  unite,  or  ones;  the  same  figure,  in  the 
second  place,  signifies  so  many  tens;  in  the  third  place, 
hundreds  ;  in  the  fourth  place,  tiiousands  ;  in  the  fifth  place, 
tens  of  thousands ;  and  in  the  sixth  place,  hundreds  of 
thousands.  Therefore  every  time  a  figure  is  removed  one 
place  to  the  left,  its  value  is  made  ten  fold,  as  may  be  seen 
in  the  following  table. 


fil 


111 


111,111,  read  One  hundred  and  eleven  thousand,  one  hundred  and  eleven, 

4    "     Four. 
49    "     Forty-nine. 
496    "     Four  hundred  and  ninety-six. 
4,965    "     Four  thousand  nine  hundred  and  sixty-five. 
49,652    "     Forty-nine  thousand,  six  hundred  and  fifty-two. 
496,527    «     Four  hundred  and  ninety-six  thousand,  five  hundred  and 
twenty-seven. 


3«     Let  the  pupil  read  the  following  numbers :  • 
5. 


34 
76 

68 
95 


142 
279 
758 
423 


9. 
10. 
11. 
12. 


6,744 
8,329 
4,960 
9,080 


13. 
14. 
15. 
16. 


1,504 

50,603 

124,769 

784,984 


4.     Let  the   pupil  express  the   following   numbers  in 
figures : — 

1.  Seventy-eight. 

2.  Ninety. 

S.     One  hundred  and  fifty-six. 
4.     Three  hundred  and  eight 


ARITHMETIC.  165 

5.  One  thousand,  four  hundred  and  sixty-nine. 

6.  Five  thousand  and  two. 

7.  Seventy-seven  thousand  and  eighty-four. 

8.  Forty  thousand,  seven  hundred  and  twenty-two. 

9.  Five  hundred  and  sixty-six  thousand,  -  one  hundred 

and  fifty. 
10.     Six  hundred  thousand. 

ADDITION. 

5.  5  and  4  and  7  are  how  many  ?     This  is  called  a 
question  in  addition.     We  put  together  or  add  the  num- 
bers, 5,  4,  and  7  to  find  their  sum  or  amount.     Thus,  the 
sum  of  5  and  4  and  7  is  16. 

Ex.  1.    What  is  the  sum  of  46  and  37  ? 
rf3.         It  is  most  convenient  to  write  the  numbers  in  order, 
£5     units  under  units,  tens  under  tens,  &c. ;  then  begin- 
ning with  the  units,  we  find  that  6  units  and  7  units 
4"      are  13  units,  or  1  ten  and  3  units.     We  write  down 
37     the  3  units  under  the  units  and  add  the  1  ten  to  the 
other  tens ;  1  (ten)  and  4  (tens)  and  3  (tens)  make  8 
83      (tens)  which  written  with  the  3,  makes  83. 

Ex.  2.    What  is  the  sum  of  736  and  875  and  394  ? 
|  Adding  the  units  we  find  6  and  5  are  11,  and  4 

|  «•  3  are  15,  or  1  ten  and  5  units.  Writing  the  5  under 
a  £  &  the  column  of  units,  we  reserve  the  1  ten  and  add 
7  3  g  it  to  the  tens  of  the  next  column  ;  thus,  1  and  3  are 
g  7  5  4,  and  7  are  11,  and  9  are  20  (tens)  or  2  hundreds, 
g  9  4  and  0  tens.  We  write  the  0  under  the  column  of 

tens,  and  add  the  2  (hundreds)  to  the  column  of 

2005  hundreds ;  thus  2  and  7  are  9,  and  8  are  1 7,  and  3 
are  20  (hundreds),  which  being  the  sum  of  the  last  column 
we  write  down.  The  answer,  therefore,  is  2005. 

6.  Add  the  following  :  — 

8.  4.  5.         6.  7.  8.  9.  10.  U. 

15         83         8         7         19         28  45  56  69 

459557  8  7  5 

12.  13.  14.  15.  16.  17.  18.  19.  20. 
35  17  75  35  62  84  93  60  80 
42  50  22  26  79  45  58  30  50 


166  ARITHMETIC. 

21  22.     23.  24.  26.      26.      27.      28. 

27  60    47  234  311     435    731    478 

32  33    55  862  525     924    311    298 

56  99    72  403  2D3     808    579    5?>7 

29.  30.  81.  32.        a3.        84. 

500  7667  7543  1812      4648      785 

314  1235  1858  1732      4620     2253 

409  5555  1776  2712      8665      226 

1492  3333  452      45 

85.  86.  87.  38.         89. 

990  8  6540  38,924  468 

6209  295  6030  67,798  26,792 

2316  17  2709  54,554  J5,864 

2201  7897  5967  33,333  80,112 


7.  40.  Add  together  39,  67,  92,  43,  84,  51. 

41.  Add  together  270,  489,  126,  46. 

42.  Add  together  2368,  235,  496,  1736. 

43.  Add  together  66,470,  801,  4009,  32,  127. 

8.  44.  A  merchant  bought  4  pieces  of  cloth ;    for  the 
first  piece  he  gave  53  dollars  ;   for  the  second,  78  dollars ; 
for   the  third,  68  dollars;    and  for  the  fourth,  67  dollars. 
How  much  did  he  give  for  the  whole  V 

45.  In  an  orchard  there  are  22  peach-trees,  95  apple- 
trees,  17  pear-trees,  and  56  plum-tivt's.      How  many  trees 
in  the  orchard  ? 

46.  A  man  bought  four  horses ;  for  the  first  he  gave  84 
dollars;  for  the  second,  150  dollars;  for  the  third,  475  dol- 
lars; and  for  the  fourth,  526  dollars.     How  much  did  he 
give  for  all  the  horses  ? 

47.  In  the  Boylston  school  there  are  810  scholars;   in 
the  Brimmer,  597  scholars;   in  the  D wight,  956  scholars; 
in  the  Elliot,  776  scholars;  in  the  Franklin,  572  scholars; 
and  in  the  Hancock,  731  scholars.     How  many  scholars  in 
these  six  schools  ? 

48.  A  gentleman,  building  a  house,  paid  the  mason  2,964 
dollars,  the  carpenter  5,723  dollars,  the  plasterer  625  dollars, 
*nd  the  painter  354  dollars.      How  much  did  the~  whole 
house  cost  V 

[Seepage  173.] 


ARITHMETIC.  167 

SUBTRACTION. 

9.  9  less  7  are  how  many  ?  This  is  called  a  question  in 
Subtraction.  We  take  away  or  subtract  7  from  9  to  find 
the  difference  or  remainder,  which  is  2. 

Ex.  1.  From  476  take  245. 

•  It  is  most  convenient  to  write  the  less  number 
|  „.  j  under  the  greater,  units  under   units,  tens  under 
|«|  tens,  etc.,  as  in  addition.     Beginning  at  the  units, 
476  we   sav  ^  from  6   leaves  1,  which  we  write  under 
245  the   units ;    then   4   (tens)   from   7  (tens)  leaves  3 
2~Q~~i  tens,  which  we  write  under  the  tens ;  and  2  (hun- 
dreds)   from   4    (hundreds)   leaves    2    (hundreds, 

which  we  write  down,  and  the  answer  is  231. 

EX.  2.  From  939  take  458. 

•*  In  this   example   we  take  8  from  9  and  there 

€  j  «  remains  1,  which  we  write  down;  but  since  we 
|||  cannot  take  5  (tens)  from  3  (tens),  we  borrow  1 
939  (hundred)  or  10  (tens)  from  the  9  (hundreds,) 
458  which,  joined  with  the  3  (tens),  makes  13  (tens)  ; 

•  then  5  (tens)  from  13  (tons)  leaves  8  (tens),  which 
we  write  down.     As  one  of  the  9  (hundreds)  has 

been  put  with  the  three  (tens),  there  remains  but  8  (hun- 
dreds). Therefore  we  say  4  (hundreds)  from  8  (hundreds) 
leaves  4  (hundreds).  This,  written  down,  makes  the  answer 
481. 

3.  4.  6.  6.  7.  8.  9. 


From 

18 

58 

35 

59 

68 

82 

95 

Take 

4 

6 

12 

83 

34 

51 

75 

Ans. 

14 

10. 

11. 

12. 

13. 

14. 

16. 

From 

285 

467 

9685 

7856 

8744 

2607 

Take 

74 

46 

5473 

5722 

4302 

1405 

16. 

17. 

18. 

19. 

20. 

21. 

•22. 

From 

12 

13 

19 

16 

24 

56 

92 

Take 

6 

5 

9 

7 

8 

7 

4 

168 


ARITHMETIC. 


From  30 
Take  20 


24. 

53 
10 


26. 
46 

27 


26. 
33 
25 


27. 
64 
29 


92 
88 


29 
84 
59 


From 
Take 


30. 

134 

25 


31. 

244 

73 


82. 
252 
171 


888 
459 


34. 

2236 
1145 


35. 

3456 
2246 


86. 

9927 
6090 


37.  From  96  take  58. 

38.  Front  4785  take  8679. 
89.  From  8344  take  5824. 
40.  From  10982  take  6470. 


41.  From  984  take  176. 

42.  From  4788  take  967. 

43.  From  7898  take  6898. 

44.  From  40816  take  5622. 


10.  45.  In  a  certain  garden  there  were  86  rose-bushes, 
but  the  worms  destroyed  83  of  them.    How  many  were  left  ? 

46.  A  man  is  66  years  of  age,  and  his  son  22.     How  old 
was  the  man  at  the  birth  of  his  son  V 

47.  A  man  gave  240  dollars  for  his  carriage,  and  121  dol- 
lars for  his  horse.      How  much  more  did  he  give  for  his 
carriage  than  for  his  horse  ? 

48.  Harry  had  84  cents  given  him  to  buy  some  marbles 
with,  but  on  his  way  to  the  store  he  lost  25  cents.     How 
many  cents  had  he  left  ? 

49.  A  girl  was  carrying  72  eggs  to  market,  but  she  let  the 
basket  fall,  and  39  eggs  were  broken.     How  many  had  she 
left? 

50.  In  an  army  there  were  14,642  men,  but  in  a  battle  789 
men  were  killed  or  wounded.     How  many  serviceable  men 
were  there  left  ? 

MULTIPLICATION. 

11.  7  times  5  are  how  many  ?     This  is  called  a  question 
in   multiplication.     To  obtain   the   answer  we  repeat  or 
wiultipiy  the  number  5,  7  times  ;  thus,  7  times  5  are  35. 

Ex.  1.  Multiply  564  by  7. 

|  Writing   the  numbers  thus,  the   7   under  the 

|  I  3  units,  we  say,  7  times  4  (units)  are  28  (units),  or 

|.g  J  2  tens  and  8  units.    We  write  down  the  8  (units) 

564  and   reserve   the   two  (tens) ;   then,   7   times  6 

7  (tens)  are  42  (tens)  and  the  2  (tens)  which  we 

8948  reserved  are  44  (tens)  or  4  hundreds  and  4  tens. 

We  write  the  4  (tens)  and  reserve  the  4  (mm- 


ARITHMETIC.  169 

dreds) ;  then,  7  times  5  (hundreds)  are  35  (hundreds), 
which,  with  the  4  (hundreds)  we  reserved  are  39  (hun- 
dreds) or  3  thousands  and  9  hundreds.  This  we  write  down, 
and  the  answer  is  3948. 

2.  3.  4.  6.  6.  7.  8. 

Multiply      8          9  13  23          43          51  65 

By  3523697 

9.  10.  11.  12.  13.  14. 

Multiply  143         424         511         821         410         821 

By  247654 

16.  16.  17.  18. 

Multiply          813  962  1243  2104 

By  3  8  2  4 

19.  20.  21.  22. 

Multiply          £-34  7110  8947  4355 

By  6  7  9  8 

23.  Multiply  129  by  35. 

1 1  Writing  the  numbers  in  order,  units  under  units, 

1 5"  «•  H  and  tens  under  tens,  we  multiply,  first,  by  the  5 
1 1 II  units,  and  write  the  result  in  its  proper  place,  as 

j  2  9     when  we  multiplied  by  one  figure.     We  multiply 

o  K     next  by  the  3  (tens)  and  write  the  first  figure  of 

the  result  in  the  place  of  tens,  because  units  mul- 

6  4  5     tiplied  by  tens  ought  to  produce  tens.     Then  add- 
?_?_!_     ing  together  the  results  obtained  by  multiplying 
4515     by  the  5  (units)  and  3  (tens),  their  sum  will  be 
4515,  which  is  the  answer. 

24.  25.          26  27.  28.  29.  30. 

Multiply      34         35         48         45         99         612         844 

By  23         25         32         44         56  54  29 

1Q2 
68 

Ans.  782 


170  ARITHMETIC. 

31.  32.          83.          34.           85.  86.           87 

Multiply      676  434      467      548       1835  2972  5963 

By    "          76  322       212      403         768  256  4678 


38.  Multiply  816  by    6. 
89.  Multiply  363  by  48. 

40.  Multiply  846  by  65. 

41.  Multiply  152  by  87. 


42.  Multiply  2523  by    47. 

43.  Multiply    682  by  746. 

44.  Multiply  8422  by  186. 

45.  Multiply  3107  by  761. 


46.  How  much  will  48  barrels  of  flour  cost  at  8  dollars  a 
barrel  ? 

47.  In  a  room  there  are  9  windows  with  16  panes  of 
glass  in  each.     How  many  panes  of  glass  in  all  the  win- 
dows ? 

48.  If  a  man  receive  47  dollars  a  month,  how  much  will 
he  receive  in  twelve  months  ? 

49.  How  much  will  126  tons  of  hay  cost,  at  13  dollars  a 
ton? 

50.  There  are  365  days  in  a  year.     How  many  days  in 
19  years? 

[See  page  175.] 

DIVISION. 

12.  24  are  how  many  times  8  ?     This  is  called  a  ques- 
tion in  division,  because  we  divide  or  separate  the  larger 
number  into  as  many  equal  parts  as  there  are  units  in  the 
smaller  number. 

13.  The  number  to  be  divided  is  called  the  dividend , 
the  number  by  which  we  divide,  the  divisor ;  and  the  result, 
the  quotient.     If  anything    is  left  after  the  division,  it  is 
called  the  remainder. 

14.  Ex.  1.  Divide  425  by  5. 

4 

in 

•,,Uo,,5)425Dmdend.  Wrfte     d°WD      the      div!(len<1      »nd 


8  5  Quotient,  divisor  5  at  the  left  of  the  dividend, 
and  draw  a  curved  line  between  them.  First  find  how 
many  times  5  is  contained  in  42  (tens).  It  is  contained  8 


ARITHMETIC.  171 

(tens)  times  with  2  (tens)  remaining.  We  write  down  the 
8  (tens)  undei  the  tens  and  suppose  the  remainder  2  (tens) 
written  at  the  left  of  the  next  figure  of  the  dividend  mak- 
ing the  number  25.  5  is  contained  in  25,  5  times  without 
remainder.  We  write  the  5  underneath  and  the  division 
is  completed. 

Division  performed  in  this  way  is  called  short  division. 

15i  Perform  in  this  way  the  following  examples : 

2.  3.  4.  5.  6.  7. 

Div*K.3)9  Divid.nd.    ?)26        8)36        4)88        5)15         8)18 

8  (luotient. 

8.  9.  10.  11.  12.  13. 

7)21  9)36  6)48  8)72  9)81  4)40 


14.                15.                  16.                  17.                  18. 
6)7-2          2)444         4)256          3)963           7)847 

19. 
9)279 

20. 
5)6825 

21. 
6)4650 

22, 
9)^72 

23. 
4)4484 

24. 
8)1312 

16.  When  the  divisor  consists  of  two  or  more  figures,  it 
is  more  convenient  to  perform  the  operation  in  the  following 
manner : 

25.  Divide  354  by  13. 

,• 

1  i  .5     S  .5 
£££    Jg 

piYisor.  13)3  5  4  (2  7  We  write  the  dividend  and  draw  a 

2  6  curved  line  on  both  sides  of  it.     We 
9  4  first  inquire  how  many  times   13   is 
9  i  contained  in  35  (tens)  the  first  two 

3  r         figures   of   the   dividend.     We   find 
that  it  is  contained  2   (tens)   times. 

Place  the  2  (tens)  at  the  right  of  the  dividend  as  the  first 
figure  of  the  quotient,  and  then  multiplying  the  divisor  13 
by  the  2  tens,  write  the  product  26  (tens)  under  the  35 
(tens)  of  the  dividend,  and  subtract.  To  the  remainder  9 


172  ARITHMETIC. 

(tens)  we  annex  4,  the  next  figure  of  the  dividend,  anC 
find  how  many  times  the  divisor  1 3  is  contained  in  94.  It 
is  contained  7  times.  Write  the  7  (units)  at  the  right  of 
the  2  (tens)  in  the  quotient  and  multiply  the  divisor  by  the 
7  (units).  Write  the  product  91  under  the  94  and  sub- 
tract, and  there  is  left  a  remainder  of  3.  13  is  therefore 
contained  in  354  27  times  with  3  remainder. 

Division  performed  in  this  way  is  called  long  division. 

IT.  Perform  in  this  way  the  following  examples : 

26.  27.  28.  29. 

13)286(22  15)235(  17)425(  13)585( 

26 
"26 
26 

8o7  81.  82.  83. 

16)1168(  13)676(  12)1728(  15)2445( 

84.  35.  36.  37 

3)301G(  17)2057(  19)2994(  15)4445( 


38.  Divide  144  by  12. 
89.  Divide  225  by  15. 
40.  Divide  180  by  12. 


41.  Divide  5456  by  22. 

42.  Divide  1411  by  36. 

43.  Divide  8844  by  22. 


44.  How  much  sugar  at  9  cents  a  pound  can  be  bought 
for  81  cents? 

45.  If  an  acre  of  land  cost  13  dollars,  how  many  acres 
can  you  buy  for  351  dollars  ? 

46.  If  a  man  can  walk  21  miles  in  a  day,  how  long  will 
it  take  him  to  walk  273  miles  ? 

47.  A  man  divided  23,796  dollars  among  his  6  children. 
How  much  did  each  receive  ? 

48.  A  railroad  company  employed  65  workmen,  and  at 
the  end  of  the  month   it  took   1755  dollars  to  pay  them. 
How  much  did  each  man  receive  ? 

49.  There  are  24  hours  in  one  day ;  how  many  days  are 
there  in  1656  hours  ? 

50.  At  97  dollars  a  ton,  how  many  tons  of  iron  may  be 
bought  for  8439  dollars. 

[See  page  175.] 


ARITHMETIC. 


173 


TABLES    OF    PROBLEMS     FOR    THE    BLACKBOARD, 


ADDITION. 


These  Tables  are  designed  to  furnish  a  number  of  problems  for  solu- 
tion on  the  Blackboard.  They  can  be  used  in  a  great  variety  of  ways  ; 
thus  in  the  table  for  Addition  the  numbers  can  be  added  as  they  stand ; 
two  or  more  can  form  a  problem  ;  the  short  columns  can  be  added,  &c. 


1. 

564  + 

731  + 

401  + 

771  + 

118 

2. 

230 

1210 

261 

3542 

1001 

3. 

1348 

8625 

4010 

300 

2071 

4. 

4689 

14807 

309 

5795 

2100 

6. 

394 

68175 

92 

7631 

7920 

6. 

6752 

8762 

5861 

387 

1927 

7. 

397 

93721 

3872 

5961 

6092 

8. 

9687 

976 

6872 

4617 

7764 

9. 

2346 

56821 

583 

3846 

389 

10. 

3741 

468 

468 

586 

9999 

11. 

287 

79276 

1684 

592 

1000 

12. 

1692 

68340 

587 

48009 

6130 

13. 

768 

6782 

4632 

3874 

5959 

14. 

483 

483 

5879 

6879 

1527 

15. 

5068 

38192 

671 

287 

8065 

16. 

473 

628 

596 

7961 

6099 

17 

786 

9172 

4869 

326 

684 

18. 

307 

791 

8734 

4891 

1888 

19. 

5961 

998 

96871 

6873 

3039 

20. 

786 

7862 

6782 

592 

5757 

21. 

597 

5917 

4839 

8783 

6495 

22. 

8643 

1010 

609 

6792 

7241 

23. 

7995 

6783 

4687 

958 

5894 

24. 

5269 

9605 

8567 

2799 

8050 

25. 

2709 

32699 

9248 

5484 

909 

26. 

9628 

6183 

5304 

7640 

9876 

27. 

6825 

8744 

9989 

2007 

74361 

28. 

9927 

4302 

5673 

8376 

67945 

29. 

6728 

5722 

3565 

9090 

46839 

80. 

2786 

7856 

4077 

7365 

59876 

174 


ARITHMETIC. 


TABLES    OF    PROBLEMS    FOR    THE    BLACKBOARD. 

SUBTRACTION. 
FIND   THE  DIFFERENCE   BETWEEN 


I, 

2. 

-  Si 

and 

u 

452 
695 

29. 
30. 

38765 
and  46872 

and 

u 

48687 
36872 

3. 
4. 

932 
684 

H 
H 

1453 
1596 

31. 
32. 

u    2983 
68741 

u 
tt 

876 
19234 

5. 
6. 

888 
1010 

M 

937 
1629 

33. 
34. 

59687 
38794 

11 
u 

3872 
68704 

7. 
8. 

554 
923 

H 

632 
451 

35. 
36. 

65823 
168721 

n 

38726 
59233 

9. 

10. 

M   909 
375 

U 
M 

1118 
4286 

37- 
38. 

59687 
38726 

u 
u 

26870 
9862 

11. 

12. 

,  2203 
4224 

U 

U 

997 
325 

39. 
40. 

40687 
56043 

u 

1134J 
687* 

13. 
14. 

u   409 
5376 

u 
H 

6102 
7635 

41. 
42. 

261807 
592687 

u 
M 

3872* 

15. 
16. 

u  6001 
3556 

u 

M 

601 
4937 

43. 

44. 

38740 
93341 

u 

u 

587? 
68743 

17. 
18. 

i  1204 
5872 

U 
U 

9005 
3478 

45. 

46. 

67215 
164327 

u 
u 

98672 
19872 

19. 

20. 

,  6226 
3248 

M 
M 

4195 
8423 

47. 
48. 

a    6824 
93678 

u 

M 

6875 
3268 

21. 
22. 

u  8049 
6412 

U 

u 

4901 
1010 

49. 
50. 

10000 
10982 

u 
u 

999S 
64  7k 

23. 
24 

;  1965 
3824 

u 
u 

3642 
4007 

51. 
52- 

79797 
40815 

u 

68888 
56225 

25. 

j  9999 
8763 

u 

u 

1098 
2854 

53. 
54. 

66476 
56324 

u 
u 

58134 
83459 

27. 

28. 

t  5542 
1537 

u 

14 

998 
3336 

55. 
56. 

789859 
984967 

a 
U 

47885 
736684 

ARITHMETIC. 


175 


TABLES    OF    PROBLEMS    FOR   THE    BLACKBOARD. 

MULTIPLICATION. 

These  numbers  are  to  be  multiplied  by  26,  37,  48,  69,  327,  468,  589,  376 
Uso  the  numbers  in  one  column  by  those  in  the  other  columns. 


1.    .37 

17.  2900 

33.  4101 

49.   6954 

65.   62437 

2.    42 

18.  3305 

34.  3762 

50.   4687 

66.   88106 

3.    56 

19.  9999 

35.  6256 

51.   6492 

67.   27091 

4.    81 

20.  5678 

36.  7815 

52.   9070 

68.   78305 

5.   253 

21.  3255 

37.  5942 

53.  46872 

69.   64092 

6.   163 

22.  2247 

38.  8307 

54.  35906 

70.   38764 

7.   813 

23.  8770 

39.  9525 

55.  43298 

71.   10001 

8.   340 

24.  4468 

40.  5015 

56.  39612 

72.   98271 

9.  2518 

25.  3173 

41.  2405 

57.  47385 

73.   64327 

10.  1130 

26.  6060 

42.  1000 

58.  96826 

74.   86475 

11.  4008 

27.  9318 

43.  1710 

59.  13478 

75.   90090 

12.  3735 

28.  2165 

44.  6341 

60.  26593 

76.  145301 

13.  4265 

29.  2924 

45.  1234 

61.  80435 

77.  309075 

14.  1704 

30.  3273 

46.  8888 

62.  72461 

78.  900000 

15.  8700 

31.  1702 

47.  1005 

63.  38675 

79.  946871 

16.  7100 

32.  5218 

48.  8095 

64.  92413 

80.  362432 

DIVISION. 

To  be  divided  by  such  numbers  as  the  Teacher  may  designate 

1.   462 

17.   847 

33.  26830:49.  90684 

65.  826541 

2.   924 

18.  8866 

34.  86109150.  43592 

66.  487455 

3.   796 

19.  4595 

35.  92165  51.  30030 

67.  305095 

4.   566 

20.  3796 

36.  42061  52.  50505 

68.  874687 

5.   248 

21.  7643 

37.  56824  53.  906870 

69.  604090 

6.   806 

22.  9262 

38.  41837  54.  765903 

70.  268405 

7.   749 

23.  2673 

39.  74587J55.  987648 

71.  368456 

8.  8181 

24.  7924 

40.  90411  ;  56.  264371 

72.  461043 

9.  4848 

25.  5050 

41.  74261|  57.  874687 

73.  782654 

10.  9635 

26.  7735 

42.  83501  58.  682409 

74.  387005 

11.  3721 

27.  26874 

43.  96843  59.  360002 

75.  100101 

12.  9900 

28.  54672 

44.  16890  60.  940856 

76.  961147 

13.  4805 

29.  87964 

45.  26874 

61.  604379 

77.  570880 

14.  4650 

30.  70865 

46.  38421  62.  200648 

78.  306890 

15.  9972 

31.  86304 

47.  92674  63.  961478 

79.  560437 

16.  6825 

32.  72591 

48.  58435 

64.  248700 

80.  818127 

176 


AMTHMETIC. 


In  the  study  of  Mental  Arithmetic,  too  much  stress  can  hardly  be  put 
upon  the  importance  of  an  intimate  familiarity  with  the  tables  of  Ad- 
dition, Subtraction,  Multiplication,  &c.  On  this  account  it  has  been 
thought  best  to  insert  here  the  Multiplication  Table,  which  may  be  re- 
ferred to  with  profit  in  the  earlier  lessons  of  the  book,  and  until,  in  all  ite 
combinations,  it  is  entirely  familiar  to  the  young  learner. 

MULTIPLICATION  TABLE. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

'  4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

To  find  how  many  8  times  6  are,  put  your  finger 
on  the  8  in  the  left-hand  column ;  then  move  the 
finger  to  the  right  till  it  comes  under  the  6  in  the 
top  line,  and  there  you  find  48,  which  is  8  times  6. 
In  the  same  manner  learn  the  whole  table. 


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edition."  —  Hartford  Press. 

A  BRIEF  BIOGRAPHICAL   DICTIONARY. 

Compiled  and  arranged  by  CHARLES  HOLE,  B.  A.,  of  Trin- 
ity College,  Cambridge ;  with  additions  and  correc- 
tions by  William  A.  Wheeler,  M.  A.,  Associate 
Editor  of  Webster's   Dictionaries.     In  one 
volume,  i6mo.  Cloth,  $1.50  ;  half 
morocco,  $3.00. 

"The  English  edition  contains  more  than  18,000  entries,  and  to 
those  Mr.  Wheeler  has  added  many  hundreds  of  American  and  a  con- 
siderable number  of  European  names  omitted  by  the  English  ecitor. 
Thus  in  the  short  space  of  five  hundred  pages  the  reader  has  the 
compressed  results  of  as  many  volumes.  The  book  takes  up  but  lit- 
tle room,  but  it  is  a  kind  of  Biographie  Universelle  to  the  reader  or 
writer  who  desires  to  carry  in  his  pocket  the  means  of  finding  or  ver- 
ifying dates/'  —  Boston  Transcript. 

PUBLISHED  BY  IIURD  AND   HOUGHTON,  NEW  YORK: 
THE   RIVERSIDE   PRESS,  CAMBRIDGE. 


